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CE72.52 - Lecture 2 - Material Behavior

Fawad Najam, profile picture
Fawad Najam

This document discusses material behavior and properties that are important for structural analysis and design. It defines various types of material stiffness, from material stiffness to cross-section stiffness to member and structure stiffness. It also discusses stress-strain relationships and different material models, including linear elastic, nonlinear elastic, plastic, and viscoelastic models. Finally, it covers key material properties like strength, stiffness, ductility, time-dependent behavior, damping properties, and how these properties depend on the material composition and loading conditions.

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1 CE 72.52 Advanced Concrete Lecture 2: Material Behavior Naveed Anwar Executive Director, AIT Consulting Director, ACECOMS Affiliate Faculty, Structural Engineering, AIT August - 2014
To Send Assignments • Sent to • javaria.aitc@ait.ac.th • Copy to • nanwar@ait.ac.th 2
Why Material Behavior is Important 3
The Structural System 4 pv Advanced Concrete l Dr. Naveed Anwar EXCITATION Loads Vibrations Settlements Thermal Changes RESPONSES Displacements Strains Stress Stress Resultants STRUCTURE
5
What is Stiffness? 6 Stiffness is the “resistance to deformation” And its opposite, Flexibility is the “ease of deformation”
What is Stiffness “made off” • The overall stiffness of the structure is derived from the overall geometry and connectivity of the members and their stiffness • The member stiffness is derived from the cross-section stiffness, and member geometry • The cross-section stiffness is derived from the material stiffness and the cross-section geometry • All of these stiffness relationships may be linear or nonlinear. 7Advanced Concrete l Dr. Naveed Anwar Material Stiffness Section Stiffness Member Stiffness Structure Stiffness Cross-Section Geometry Member Geometry Structure Geometry
What is Stiffness “made off” • The overall resistance of the structures to overall loads, called the Global Structure Stiffness. • This is derived from the sum of stiffness of its members, their connectivity and the boundary or the restraining conditions. • The resistance of each member to local actions called the Member Stiffness is derived from the cross-section stiffness and the geometry of the member. • The resistance of the cross-section to overall strains. This is derived from the cross-section geometry and the stiffness of the materials from which it is made. • The resistance of the material to strain derived from the stiffness of the material particles. 8
The Response and Design 9 Advanced Concrete l Dr. Naveed Anwar
Loads and Stress Resultants 10 Advanced Concrete l Dr. Naveed Anwar Depends on K Depends on K
Structural Materials 11
The Structural Materials • By “structural material”, we mean the material for which mechanical properties are usually defined for the purpose of structural analysis and design. 12Advanced Concrete l Dr. Naveed Anwar
Material Hierarchy 13
Structural Materials Concrete Plain Reinforced Pre-stressed Confined/un-confined Fercoement Metals Steel Aluminum Non-Metals Carbon/Glass fibers Wood 14
Material Behavior and Properties 15
Stress and Strain • The Hook's law states that within the elastic limits, the stress is proportional to the strain • This is valid for only Very Limited cases • Modulus of Elasticity, E is NOT a constant • There are many stress and strain components, and many properties 16Advanced Concrete l Dr. Naveed Anwar Strain Stress E (mod)
A Bigger Picture of Stress-Strain Components 17 xx yy zz xy zx yx zy xz yz x y z At any point in a continuum, or solid, the stress state can be completely defined in terms of six stress components and six corresponding strains. xx yy zz xy zx yx zy xz yz x y z At any point in a continuum, or solid, the stress state can be completely defined in terms of six stress components and six corresponding strains.
Typical Force-Displacement or Stress-Strain 18
Basic Properties • For analysis • Modulus of elasticity, E • Poisons ratio, mu • Shear modulus, G • Thermal expansion coefficient, alpha • For design • Yield stress, Fy • Failure stress, Fu, Fc, Ft etc. • Yield strain • Failure strain 19
Other Specific properties • Relaxation • Fatigue • Creep • Shrinkage • Confinement based 20
Dependence of Behavior • Relationship between Stress and Strain Depends on • Basic material composition • Initial conditions • State of strain • Direction of strain • History of strain • Time since initial strain • Temperature • Cyclic strain • Rate of strain change, Velocity and acceleration 21
Linearity and Elasticity • Material behavior depends on level of strain • Linear • Non-linear • Material behavior depends on loading history • Elastic • Plastic • Inelastic • Hysteretic 22
Linear Elastic Material • A linear elastic material is one in which the strain is proportional to stress • Both “loading” and “unloading” curves are same (straight lines). 23Advanced Concrete l Dr. Naveed Anwar Strain Stress
Linear Inelastic Material • A linear inelastic material is one in which the strain is proportional to stress • “Loading” and “unloading” curves are not same (although straight lines). 24 Strain Stress Advanced Concrete l Dr. Naveed Anwar
Nonlinear Elastic material • For a nonlinear elastic material, strain is not proportional to stress as shown in figure. • Both “loading” and “unloading” curves are same but are not straight lines. 25 Strain Stress Advanced Concrete l Dr. Naveed Anwar
Nonlinear Inelastic Material • For a nonlinear inelastic material, strain is not proportional to stress as shown in figure. • “Loading” and “unloading” curves are not same in this case. 26 Strain Stress Advanced Concrete l Dr. Naveed Anwar
Elastic–Perfectly Plastic (Non-strain Hardening) • The behavior of an elastic-perfectly plastic (non-strain hardening) material 27 Strain Stress Advanced Concrete l Dr. Naveed Anwar
Elastic – Plastic Material • The elastic plastic material exhibits a stress – strain behavior as depicted in the figure 28 Strain Stress Advanced Concrete l Dr. Naveed Anwar
Ductile and Brittle Materials • Ductile materials: • able to deform significantly into the inelastic range • Brittle materials: • fail suddenly by cracking or splintering • much weaker in tension than in compression 29Advanced Concrete l Dr. Naveed Anwar Deformation Force ductile Deformation Force brittle
A Bigger Picture of Stress-Strain Components 30 xx yy zz xy zx yx zy xz yz x y z At any point in a continuum, or solid, the stress state can be completely defined in terms of six stress components and six corresponding strains. xx yy zz xy zx yx zy xz yz x y z At any point in a continuum, or solid, the stress state can be completely defined in terms of six stress components and six corresponding strains.                                                                              zx yz xy z y x zx yz xy z y x v v v vvv vvv vvv vv E             2 21 00000 0 2 21 0000 00 2 21 000 0001 0001 0001 211
Basic Directional Material Behavior • Isotropic • Same behavior in all directions • All directions un-coupled • Orthotropic • Different behavior in orthogonal directions • Behavior is un-coupled • Anisotropic • Different behavior in 3 directions • Behavior is coupled 31
Simplified Case of Beam Section – Isotropic Case 32     yyxxx vvv E   11 1 2   xyxy E v    12 Replacing  v E G   12 G xy xy    E xx x    For beam cross-sections, we can neglect yy and the squares of v. We then get the simple relationships between stress and strain involving only E and G. The full relationship can be simplified for a beam type member where only three stresses and strains are of importance                                                                              zx yz xy z y x zx yz xy z y x v v v vvv vvv vvv vv E             2 21 00000 0 2 21 0000 00 2 21 000 0001 0001 0001 211
Stiffness Component 33 Deformation Force Curvature Moment Section Stiffness Member Stiffness Structure Stiffness Material Stiffness Structure Geometry Member Geometry Cross-section Geometry Rotation Moment Strain Stress
Basic Properties 34
Orthotropic 35
Anisotropic 36
Uniaxial 37
Confinement Properties 38
Nonlinear Data 39
Damping Properties 40
Time Dependent Properties 41
Strength 42
Strength, Stiffness and Ductility • Strength (ultimate stress): the stress (load per unit area of the cross-section) at which the failure takes place • tension • Compression • Stiffness: the resistance of an elastic body to deformation • Ductility: capacity of the material to deform into the inelastic range without significant loss of its load-bearing capacity 43Advanced Concrete l Dr. Naveed Anwar
The Concept of Specific Strength • First to realize this: Galileo Galilei (1654AD – 1642 AD) • All structures have to support their own weight • Can the size of a structure be increased indefinitely for it to be able to carry its own weight? • Problem: how long a bar of uniform cross-section can be before it breaks due to its own weight? • Equate the weight of the bar to its tensile strength: • Weight = Tensile resistance 44 Advanced Concrete l Dr. Naveed Anwar Cross-sectional area A L
The Concept of Specific Strength • Weight = Volume × specific weight • W = A × L × ρ × g • Tensile resistance = Area × Ultimate tensile strength • R = A × Tu • Equate weight to resistance: • W = R A × L × ρ × g = A × Tu • L = Tu / (r × g) = S = specific strength • There is an absolute limit (= S) to the length that the bar can attain without breaking • Larger a structure is, larger is the proportion of its own weight to the total load that can be carried by itself 45 Cross-sectional area A L Advanced Concrete l Dr. Naveed Anwar
The Concept of Specific Strength • For structures subjected to tension/compression, as the size of an object increases, its strength increases with the square of the ruling dimensions, while the weight increases with its cube • For each type of structure there is a maximum possible size beyond which it cannot carry even its own weight • Consequences: • it is impossible to construct structures of enormous size • there is a limit to natural structures (trees, animals, etc.) • larger a structure becomes, stockier and more bulky it gets • large bridges are heavier in proportions than smaller ones • bones of elephants are stockier and thicker than the ones of mice • proportions of aquatic animals are almost unaffected by their size (weight is almost entirely supported by buoyancy) 46 Advanced Concrete l Dr. Naveed Anwar
Ultimate and Specific Strengths 47Advanced Concrete l Dr. Naveed Anwar
Specific Strength • Stone, brick and concrete: used in compression • Steel: used in tension • Timber: excellent performance in terms of specific strength, especially in tension • Aluminum: high specific strength • Aircrafts must carry loads and must be capable of being raised into the air under their own power materials with high specific strength • wood was extensively used in early planes • modern material: aluminium 48Advanced Concrete l Dr. Naveed Anwar
Structural materials: Stiffness and Ductility 49Advanced Concrete l Dr. Naveed Anwar
Structural materials: Ductility • Ductility is important for the "ultimate" behavior of structures • Most structures are designed to respond in the elastic range under service loads, but, given the uncertainties in real strength of material, behavior of the structure, magnitude of loading, and accidental actions, a structure can be subjected to inelastic deformations 50 Advanced Concrete l Dr. Naveed Anwar
Structural materials: Ductility • A ductile material will sustain large deformations before collapsing, "warning" the people inside • A ductile material allows for redistribution of stresses in statically indeterminate structures, which are able to support larger loads than in the case of a structure realized of brittle material 51
Stress Strain Relationships for Structural Concrete and Steel 52
Concrete Stress Strain Curve- BS8110 53 0.0035 0.67 fu/ γm Stress Strain 2.4 x 10-4 (fcu/γm)1/2 5.5 (fcu/γm)1/2 Advanced Concrete l Dr. Naveed Anwar
Various Concrete Models 54 Advanced Concrete l Dr. Naveed Anwar
Various Concrete Models 55 Linear Whitney PCA BS-8110 Parabolic Unconfined Mander-1 Mander-2 Strain Stress Advanced Concrete l Dr. Naveed Anwar Concrete Stress-Strain Relationships
Material Ductility - Steel • Various Stress-Strain Curves for Steel reinforcement and steel sections   y h su syf suf Parabola   y h su syf suf Parabola   syf y su  syf y su   y h su suf   y h su suf   y h su syf suf Parabola   y h su syf suf Parabola Various Stress-Strain Curves for Steel reinforcement and steel sections.
Steel Reinforcement Behavior
Material Ductility - Concrete • Stress- Strain Relation as given in British code           cc cu ccf  cuf    cc cu ccf  cuf  Stress-Strain Relation for Confined Concrete Stress-Strain Relation for Concrete after Whitney cf  uf  Stress-Strain Relation as given in British code General Stress Strain curve   Stress-Strain Relation for Un Confined Concrete cc cf    Stress-Strain Relation for Un Confined Concrete cc cf    0.0035 m cuf  4 104.2   m cuf  67.0   0.0035 m cuf  4 104.2   m cuf  67.0 cf 85.0
Concrete Behavior and Confinement • Unconfined Concrete Stress-Strain Behavior
Concrete Behavior and Confinement • Idealized Stress-Strain Behavior of Unconfined Concrete
Confinements
Concrete Behavior and Confinement • Confined Concrete Stress-Strain Behavior
Concrete Behavior and Confinement • Idealized Stress-Strain Behavior of Confined Concrete
Comparison of Confine and Un-Confined Concrete • Unconfined Concrete Stress-Strain Behavior • Confined Concrete Stress-Strain Behavior
Mander’s Model for Unconfined Concrete 65                              nk c cf c cf cc n n ff ' ' ' 1     ccf   17 8.0 ' cf n                1 ' ' n n E f c c c 69003320 '  cc fE 62 67.0 ' cf k  fc’= unconfined compressive strength of concrete ’c =strain due f’c (MPa) cf = final concrete strain n = modular ratio (MPa) Ec = initial tangent stiffness of the concrete k = post-peak decay factor (MPa) which value must not be less than unity. fc= stress in concrete at any level [Mander et al. and Collins and Mitchell (1991)] Advanced Concrete l Dr. Naveed Anwar
Mander’s Model for Confined Concrete 66 concconc fff 1.4'' ,  = confining pressure [Collins and Mitchell (1991)] Advanced Concrete l Dr. Naveed Anwar
Cyclic Stress Strain Relationship for Concrete 67 Reference: James G. Macgregor Reinforced Concrete: Mechanics and Design, 3rd Edition Advanced Concrete l Dr. Naveed Anwar
Steel Stress Strain Curve- BS8110 68 fy/ γm fy/ γm Stress Strain Compression Tension 200 kN/mm2 Advanced Concrete l Dr. Naveed Anwar
Various Steel Models 69 Strain Stress Linear - Elastic Elasto-Plastic Strain Hardening - Simple Strain Hardening Park Advanced Concrete l Dr. Naveed Anwar Steel Stress-Strain Relationships
Steel Stress Strain Relationship 70 Steel: Stress-strain diagrams for different steels (Hibbeler, 1997) Reference: James G. Macgregor Reinforced Concrete: Mechanics and Design, 3rd Edition Advanced Concrete l Dr. Naveed Anwar
Steel Stress Strain Relationship 71 Reference: James G. Macgregor Reinforced Concrete: Mechanics and Design, 3rd Edition Advanced Concrete l Dr. Naveed Anwar
Steel Stress Strain Relationship • The reinforcing steel is assumed to be elastic until the yield strain y, and perfectly plastic for strains between y and the hardening strain or until the limit strain su, represented by three linear relationship. However, two linear stress-strain relationship is still being used, and simply expressed as • fs= stress in reinforcing steel at any level y due to s • Es = 200,000 MPa , modulus of elasticity of reinforcing steel 72 sss Ef  ys  , for ys ff  ys  , if Advanced Concrete l Dr. Naveed Anwar
Steel strain due to thermal changes • At low temperatures, steel becomes harder and more brittle while it becomes softer and more ductile when the temperature rises. Although the thermal expansion for steel is actually 6.5x10-6/0F, it is conventional to use a value of 6x10-6/0F for both concrete and reinforcement (Collins and Mitchell, 1991) 73 Tsbs   sb Fx 06 /106  , coefficient of thermal expansion T = changes in temperature in 0F Advanced Concrete l Dr. Naveed Anwar
Assignment - 2 • Summarize 5 Concrete Confinement Models, with equations and compare their stress strain curve for a column • 500x500, fc= 40 MPA, confined by hoops of dia 12 @ 200 mm, and 16 vertical bars of dia 25 mm • Time 1 week 74
Standard Reinforcing Bars (US Designation) 75 Nominal Dimensions* Bar Size Designation No. Grades Weight (lb/ft) Diameter (in.) Cross-Sectional Area (in2 .) 3 40, 60 0.376 0.375 0.11 4 40, 60 0.668 0.500 0.20 5 40, 60 1.043 0.625 0.31 6 40, 60 1.502 0.750 0.44 7 60 2.044 0.875 0.60 8 60 2.670 1.000 0.79 9 60 3.40 1.128 1.00 10 60 4.30 1.270 1.27 11 60 5.31 1.410 1.56 14 60 7.65 1.693 2.25 18 60 13.60 2.257 4.00 Advanced Concrete l Dr. Naveed Anwar * The nominal dimensions of a deformed bars are equivalent to those of a plan round bar having the same weight per foot as the deformed bar.
76

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CE72.52 - Lecture 2 - Material Behavior

  • 1. 1 CE 72.52 Advanced Concrete Lecture 2: Material Behavior Naveed Anwar Executive Director, AIT Consulting Director, ACECOMS Affiliate Faculty, Structural Engineering, AIT August - 2014
  • 2. To Send Assignments • Sent to • javaria.aitc@ait.ac.th • Copy to • nanwar@ait.ac.th 2
  • 4. The Structural System 4 pv Advanced Concrete l Dr. Naveed Anwar EXCITATION Loads Vibrations Settlements Thermal Changes RESPONSES Displacements Strains Stress Stress Resultants STRUCTURE
  • 5. 5
  • 6. What is Stiffness? 6 Stiffness is the “resistance to deformation” And its opposite, Flexibility is the “ease of deformation”
  • 7. What is Stiffness “made off” • The overall stiffness of the structure is derived from the overall geometry and connectivity of the members and their stiffness • The member stiffness is derived from the cross-section stiffness, and member geometry • The cross-section stiffness is derived from the material stiffness and the cross-section geometry • All of these stiffness relationships may be linear or nonlinear. 7Advanced Concrete l Dr. Naveed Anwar Material Stiffness Section Stiffness Member Stiffness Structure Stiffness Cross-Section Geometry Member Geometry Structure Geometry
  • 8. What is Stiffness “made off” • The overall resistance of the structures to overall loads, called the Global Structure Stiffness. • This is derived from the sum of stiffness of its members, their connectivity and the boundary or the restraining conditions. • The resistance of each member to local actions called the Member Stiffness is derived from the cross-section stiffness and the geometry of the member. • The resistance of the cross-section to overall strains. This is derived from the cross-section geometry and the stiffness of the materials from which it is made. • The resistance of the material to strain derived from the stiffness of the material particles. 8
  • 9. The Response and Design 9 Advanced Concrete l Dr. Naveed Anwar
  • 10. Loads and Stress Resultants 10 Advanced Concrete l Dr. Naveed Anwar Depends on K Depends on K
  • 12. The Structural Materials • By “structural material”, we mean the material for which mechanical properties are usually defined for the purpose of structural analysis and design. 12Advanced Concrete l Dr. Naveed Anwar
  • 16. Stress and Strain • The Hook's law states that within the elastic limits, the stress is proportional to the strain • This is valid for only Very Limited cases • Modulus of Elasticity, E is NOT a constant • There are many stress and strain components, and many properties 16Advanced Concrete l Dr. Naveed Anwar Strain Stress E (mod)
  • 17. A Bigger Picture of Stress-Strain Components 17 xx yy zz xy zx yx zy xz yz x y z At any point in a continuum, or solid, the stress state can be completely defined in terms of six stress components and six corresponding strains. xx yy zz xy zx yx zy xz yz x y z At any point in a continuum, or solid, the stress state can be completely defined in terms of six stress components and six corresponding strains.
  • 19. Basic Properties • For analysis • Modulus of elasticity, E • Poisons ratio, mu • Shear modulus, G • Thermal expansion coefficient, alpha • For design • Yield stress, Fy • Failure stress, Fu, Fc, Ft etc. • Yield strain • Failure strain 19
  • 20. Other Specific properties • Relaxation • Fatigue • Creep • Shrinkage • Confinement based 20
  • 21. Dependence of Behavior • Relationship between Stress and Strain Depends on • Basic material composition • Initial conditions • State of strain • Direction of strain • History of strain • Time since initial strain • Temperature • Cyclic strain • Rate of strain change, Velocity and acceleration 21
  • 22. Linearity and Elasticity • Material behavior depends on level of strain • Linear • Non-linear • Material behavior depends on loading history • Elastic • Plastic • Inelastic • Hysteretic 22
  • 23. Linear Elastic Material • A linear elastic material is one in which the strain is proportional to stress • Both “loading” and “unloading” curves are same (straight lines). 23Advanced Concrete l Dr. Naveed Anwar Strain Stress
  • 24. Linear Inelastic Material • A linear inelastic material is one in which the strain is proportional to stress • “Loading” and “unloading” curves are not same (although straight lines). 24 Strain Stress Advanced Concrete l Dr. Naveed Anwar
  • 25. Nonlinear Elastic material • For a nonlinear elastic material, strain is not proportional to stress as shown in figure. • Both “loading” and “unloading” curves are same but are not straight lines. 25 Strain Stress Advanced Concrete l Dr. Naveed Anwar
  • 26. Nonlinear Inelastic Material • For a nonlinear inelastic material, strain is not proportional to stress as shown in figure. • “Loading” and “unloading” curves are not same in this case. 26 Strain Stress Advanced Concrete l Dr. Naveed Anwar
  • 27. Elastic–Perfectly Plastic (Non-strain Hardening) • The behavior of an elastic-perfectly plastic (non-strain hardening) material 27 Strain Stress Advanced Concrete l Dr. Naveed Anwar
  • 28. Elastic – Plastic Material • The elastic plastic material exhibits a stress – strain behavior as depicted in the figure 28 Strain Stress Advanced Concrete l Dr. Naveed Anwar
  • 29. Ductile and Brittle Materials • Ductile materials: • able to deform significantly into the inelastic range • Brittle materials: • fail suddenly by cracking or splintering • much weaker in tension than in compression 29Advanced Concrete l Dr. Naveed Anwar Deformation Force ductile Deformation Force brittle
  • 30. A Bigger Picture of Stress-Strain Components 30 xx yy zz xy zx yx zy xz yz x y z At any point in a continuum, or solid, the stress state can be completely defined in terms of six stress components and six corresponding strains. xx yy zz xy zx yx zy xz yz x y z At any point in a continuum, or solid, the stress state can be completely defined in terms of six stress components and six corresponding strains.                                                                              zx yz xy z y x zx yz xy z y x v v v vvv vvv vvv vv E             2 21 00000 0 2 21 0000 00 2 21 000 0001 0001 0001 211
  • 31. Basic Directional Material Behavior • Isotropic • Same behavior in all directions • All directions un-coupled • Orthotropic • Different behavior in orthogonal directions • Behavior is un-coupled • Anisotropic • Different behavior in 3 directions • Behavior is coupled 31
  • 32. Simplified Case of Beam Section – Isotropic Case 32     yyxxx vvv E   11 1 2   xyxy E v    12 Replacing  v E G   12 G xy xy    E xx x    For beam cross-sections, we can neglect yy and the squares of v. We then get the simple relationships between stress and strain involving only E and G. The full relationship can be simplified for a beam type member where only three stresses and strains are of importance                                                                              zx yz xy z y x zx yz xy z y x v v v vvv vvv vvv vv E             2 21 00000 0 2 21 0000 00 2 21 000 0001 0001 0001 211
  • 33. Stiffness Component 33 Deformation Force Curvature Moment Section Stiffness Member Stiffness Structure Stiffness Material Stiffness Structure Geometry Member Geometry Cross-section Geometry Rotation Moment Strain Stress
  • 43. Strength, Stiffness and Ductility • Strength (ultimate stress): the stress (load per unit area of the cross-section) at which the failure takes place • tension • Compression • Stiffness: the resistance of an elastic body to deformation • Ductility: capacity of the material to deform into the inelastic range without significant loss of its load-bearing capacity 43Advanced Concrete l Dr. Naveed Anwar
  • 44. The Concept of Specific Strength • First to realize this: Galileo Galilei (1654AD – 1642 AD) • All structures have to support their own weight • Can the size of a structure be increased indefinitely for it to be able to carry its own weight? • Problem: how long a bar of uniform cross-section can be before it breaks due to its own weight? • Equate the weight of the bar to its tensile strength: • Weight = Tensile resistance 44 Advanced Concrete l Dr. Naveed Anwar Cross-sectional area A L
  • 45. The Concept of Specific Strength • Weight = Volume × specific weight • W = A × L × ρ × g • Tensile resistance = Area × Ultimate tensile strength • R = A × Tu • Equate weight to resistance: • W = R A × L × ρ × g = A × Tu • L = Tu / (r × g) = S = specific strength • There is an absolute limit (= S) to the length that the bar can attain without breaking • Larger a structure is, larger is the proportion of its own weight to the total load that can be carried by itself 45 Cross-sectional area A L Advanced Concrete l Dr. Naveed Anwar
  • 46. The Concept of Specific Strength • For structures subjected to tension/compression, as the size of an object increases, its strength increases with the square of the ruling dimensions, while the weight increases with its cube • For each type of structure there is a maximum possible size beyond which it cannot carry even its own weight • Consequences: • it is impossible to construct structures of enormous size • there is a limit to natural structures (trees, animals, etc.) • larger a structure becomes, stockier and more bulky it gets • large bridges are heavier in proportions than smaller ones • bones of elephants are stockier and thicker than the ones of mice • proportions of aquatic animals are almost unaffected by their size (weight is almost entirely supported by buoyancy) 46 Advanced Concrete l Dr. Naveed Anwar
  • 47. Ultimate and Specific Strengths 47Advanced Concrete l Dr. Naveed Anwar
  • 48. Specific Strength • Stone, brick and concrete: used in compression • Steel: used in tension • Timber: excellent performance in terms of specific strength, especially in tension • Aluminum: high specific strength • Aircrafts must carry loads and must be capable of being raised into the air under their own power materials with high specific strength • wood was extensively used in early planes • modern material: aluminium 48Advanced Concrete l Dr. Naveed Anwar
  • 49. Structural materials: Stiffness and Ductility 49Advanced Concrete l Dr. Naveed Anwar
  • 50. Structural materials: Ductility • Ductility is important for the "ultimate" behavior of structures • Most structures are designed to respond in the elastic range under service loads, but, given the uncertainties in real strength of material, behavior of the structure, magnitude of loading, and accidental actions, a structure can be subjected to inelastic deformations 50 Advanced Concrete l Dr. Naveed Anwar
  • 51. Structural materials: Ductility • A ductile material will sustain large deformations before collapsing, "warning" the people inside • A ductile material allows for redistribution of stresses in statically indeterminate structures, which are able to support larger loads than in the case of a structure realized of brittle material 51
  • 52. Stress Strain Relationships for Structural Concrete and Steel 52
  • 53. Concrete Stress Strain Curve- BS8110 53 0.0035 0.67 fu/ γm Stress Strain 2.4 x 10-4 (fcu/γm)1/2 5.5 (fcu/γm)1/2 Advanced Concrete l Dr. Naveed Anwar
  • 54. Various Concrete Models 54 Advanced Concrete l Dr. Naveed Anwar
  • 55. Various Concrete Models 55 Linear Whitney PCA BS-8110 Parabolic Unconfined Mander-1 Mander-2 Strain Stress Advanced Concrete l Dr. Naveed Anwar Concrete Stress-Strain Relationships
  • 56. Material Ductility - Steel • Various Stress-Strain Curves for Steel reinforcement and steel sections   y h su syf suf Parabola   y h su syf suf Parabola   syf y su  syf y su   y h su suf   y h su suf   y h su syf suf Parabola   y h su syf suf Parabola Various Stress-Strain Curves for Steel reinforcement and steel sections.
  • 58. Material Ductility - Concrete • Stress- Strain Relation as given in British code           cc cu ccf  cuf    cc cu ccf  cuf  Stress-Strain Relation for Confined Concrete Stress-Strain Relation for Concrete after Whitney cf  uf  Stress-Strain Relation as given in British code General Stress Strain curve   Stress-Strain Relation for Un Confined Concrete cc cf    Stress-Strain Relation for Un Confined Concrete cc cf    0.0035 m cuf  4 104.2   m cuf  67.0   0.0035 m cuf  4 104.2   m cuf  67.0 cf 85.0
  • 59. Concrete Behavior and Confinement • Unconfined Concrete Stress-Strain Behavior
  • 60. Concrete Behavior and Confinement • Idealized Stress-Strain Behavior of Unconfined Concrete
  • 62. Concrete Behavior and Confinement • Confined Concrete Stress-Strain Behavior
  • 63. Concrete Behavior and Confinement • Idealized Stress-Strain Behavior of Confined Concrete
  • 64. Comparison of Confine and Un-Confined Concrete • Unconfined Concrete Stress-Strain Behavior • Confined Concrete Stress-Strain Behavior
  • 65. Mander’s Model for Unconfined Concrete 65                              nk c cf c cf cc n n ff ' ' ' 1     ccf   17 8.0 ' cf n                1 ' ' n n E f c c c 69003320 '  cc fE 62 67.0 ' cf k  fc’= unconfined compressive strength of concrete ’c =strain due f’c (MPa) cf = final concrete strain n = modular ratio (MPa) Ec = initial tangent stiffness of the concrete k = post-peak decay factor (MPa) which value must not be less than unity. fc= stress in concrete at any level [Mander et al. and Collins and Mitchell (1991)] Advanced Concrete l Dr. Naveed Anwar
  • 66. Mander’s Model for Confined Concrete 66 concconc fff 1.4'' ,  = confining pressure [Collins and Mitchell (1991)] Advanced Concrete l Dr. Naveed Anwar
  • 67. Cyclic Stress Strain Relationship for Concrete 67 Reference: James G. Macgregor Reinforced Concrete: Mechanics and Design, 3rd Edition Advanced Concrete l Dr. Naveed Anwar
  • 68. Steel Stress Strain Curve- BS8110 68 fy/ γm fy/ γm Stress Strain Compression Tension 200 kN/mm2 Advanced Concrete l Dr. Naveed Anwar
  • 69. Various Steel Models 69 Strain Stress Linear - Elastic Elasto-Plastic Strain Hardening - Simple Strain Hardening Park Advanced Concrete l Dr. Naveed Anwar Steel Stress-Strain Relationships
  • 70. Steel Stress Strain Relationship 70 Steel: Stress-strain diagrams for different steels (Hibbeler, 1997) Reference: James G. Macgregor Reinforced Concrete: Mechanics and Design, 3rd Edition Advanced Concrete l Dr. Naveed Anwar
  • 71. Steel Stress Strain Relationship 71 Reference: James G. Macgregor Reinforced Concrete: Mechanics and Design, 3rd Edition Advanced Concrete l Dr. Naveed Anwar
  • 72. Steel Stress Strain Relationship • The reinforcing steel is assumed to be elastic until the yield strain y, and perfectly plastic for strains between y and the hardening strain or until the limit strain su, represented by three linear relationship. However, two linear stress-strain relationship is still being used, and simply expressed as • fs= stress in reinforcing steel at any level y due to s • Es = 200,000 MPa , modulus of elasticity of reinforcing steel 72 sss Ef  ys  , for ys ff  ys  , if Advanced Concrete l Dr. Naveed Anwar
  • 73. Steel strain due to thermal changes • At low temperatures, steel becomes harder and more brittle while it becomes softer and more ductile when the temperature rises. Although the thermal expansion for steel is actually 6.5x10-6/0F, it is conventional to use a value of 6x10-6/0F for both concrete and reinforcement (Collins and Mitchell, 1991) 73 Tsbs   sb Fx 06 /106  , coefficient of thermal expansion T = changes in temperature in 0F Advanced Concrete l Dr. Naveed Anwar
  • 74. Assignment - 2 • Summarize 5 Concrete Confinement Models, with equations and compare their stress strain curve for a column • 500x500, fc= 40 MPA, confined by hoops of dia 12 @ 200 mm, and 16 vertical bars of dia 25 mm • Time 1 week 74
  • 75. Standard Reinforcing Bars (US Designation) 75 Nominal Dimensions* Bar Size Designation No. Grades Weight (lb/ft) Diameter (in.) Cross-Sectional Area (in2 .) 3 40, 60 0.376 0.375 0.11 4 40, 60 0.668 0.500 0.20 5 40, 60 1.043 0.625 0.31 6 40, 60 1.502 0.750 0.44 7 60 2.044 0.875 0.60 8 60 2.670 1.000 0.79 9 60 3.40 1.128 1.00 10 60 4.30 1.270 1.27 11 60 5.31 1.410 1.56 14 60 7.65 1.693 2.25 18 60 13.60 2.257 4.00 Advanced Concrete l Dr. Naveed Anwar * The nominal dimensions of a deformed bars are equivalent to those of a plan round bar having the same weight per foot as the deformed bar.
  • 76. 76