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
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.
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
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
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
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
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
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.