CE 412 Lecture 2 Material Behaviour
CE 412 Lecture 2 Material Behaviour
CE 412 Lecture 2 Material Behaviour
Lecture 2:
Materials
October - 2020
2
Stresses and Strains – General Notation
• The Hook's law states that within the elastic limits, the stress is proportional to the strain
𝜎=𝐸𝜀
• There are many stress and strain components, and many properties.
y yy
x
yz yx
z
•
zy xy The Hook's law is a simplified form of Stress-Strain
relationship.
xx
zx xz
zz
• Ultimately, the six stress and strain components can be
The concrete which contains steel bars to enhance its tensile strength
• Greater resistance to fire and weathering effects • Forms (shuttering) are required to hold in place
• Higher stiffness • Low weight per unit volume causes heavier structural
members
• Low maintenance material
• Variation in properties through a member.
• Increase in strength with passage of time
• Is very sensitive to hydration – related to strength
• Cost effective
• Both materials are managing the pros and cons of each other like:
• Compressive strength
• The strength may vary from 3000 to 6000 psi as normal concrete and beyond 6000 psi as high strength
concrete
• Tensile strength
• Shear Strength
• Sizes start from #3 to #11 and for special cases we may refer
to #14 and #18 also
• Shear modulus, 𝐺
• Thermal expansion coefficient, 𝛼
• For design
• Yield stress, 𝐹𝑦
• Yield strain 𝜀𝑦
• Failure strain 𝜀𝑢
• Relaxation
• Fatigue
• Creep
• Shrinkage
• Problem: how long a bar of uniform cross-section can be before it breaks Cross-sectional
due to its own weight? area A L
• Equate the weight of the bar to its tensile strength:
• Weight = Tensile resistance
Cross-sectional
• There is an absolute limit (= S) to the length that the bar can attain without breaking area A
L
• Larger a structure is, larger is the proportion of its own weight to the total load that can be
carried by itself
• 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).
• 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: aluminum
Isotropic Materials
• The behavior of an isotropic material is independent of the direction of loading or the orientation of the material.
• Shearing behavior is uncoupled from ex-tensional behavior and is not affected by temperature change.
Orthotropic Materials
• The behavior of an orthotropic material can be different in each of the three local coordinate directions.
• Shearing behavior is un-coupled from extensional behavior and is not affected by temperature change.
Anisotropic Materials
• The behavior of an anisotropic material can be different in each of the three local coordinate directions
• Shearing behavior can be fully coupled with extensional behavior and can be affected by temperature change.
Isotropic
Material
• Non-linear
• Plastic
• Inelastic
• Hysteretic
• A linear elastic material is one in which the • A linear inelastic material is one in which
strain is proportional to stress the strain is proportional to stress
• Both “loading” and “unloading” curves are • “Loading” and “unloading” curves are not
same (straight lines). same (although straight lines).
• For a nonlinear elastic material, strain is not • For a nonlinear inelastic material, strain is
proportional to stress as shown in figure. not proportional to stress as shown in figure.
• Both “loading” and “unloading” curves are same • “Loading” and “unloading” curves are not
but are not straight lines. same in this case.
Stress
Strain Strain
• History of strain
30
Typical Stress-strain Curves
plastic
34
Various stress-strain Models for Unconfined Concrete
Force
• Ductile materials:
ductile
• able to deform significantly into the inelastic range
• Brittle materials:
• fail suddenly by cracking or splintering
Deformation
• much weaker in tension than in compression
Force
brittle
Deformation
• 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.
• 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.
cf
n ' 𝑓𝑐 = stress in concrete at any level
' c
fc fc
n 1 cf
nk
𝑓𝑐 ’ = unconfined compressive strength of concrete
'
c ’𝑐 = strain at f’c
17 62
Reading Assignment 1
Reading Assignment 2
56
Steel Reinforcement Behavior
Stress
𝒇𝒚 /𝜸𝒎 Tension
Strain
𝝐𝒚
Compression 𝒇𝒚 /𝜸𝒎
Stress
Strain
• The reinforcing steel is assumed to be elastic until the yield strain 𝜀𝑦 , and perfectly plastic for
strains between 𝜀𝑦 and the hardening strain or until the limit strain 𝜀𝑠𝑢 , represented by three
linear relationship. However, two linear stress-strain relationship is still being used, and simply
expressed as
f s E s s , for s y
fs f y , if s y
Nominal Dimensions*
* 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.
• 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)
s sb T
sb = 6 x106 / 0 F , coefficient of thermal expansion
T = changes in temperature in 0F
• Properties at other temperatures are obtained by linear interpolation between the two nearest
specified temperatures.
• Properties at temperatures outside the specified range use the properties at the nearest specified
temperature.
66
Temperature Dependent Properties
• Fiber Hinges
• Layered Shell Element
1) Summarize 5 Models for unconfined concrete, with equations and compare their stress-strain curve.
Develop a computer program for the parametric analysis and graphical comparison of these models.
2) Summarize 5 Models for confined concrete, with equations and compare their stress-strain curve for a
column with following details.
[500 x 500 mm, 𝑓𝑐′ = 40 MPa, 16 vertical bars of dia. 25 mm (𝑓𝑦 = 420 MPa), and confined by hoops of dia. 12 @
200 mm (𝑓𝑦 = 420 MPA)]
Develop a computer program for the parametric analysis and graphical comparison of these models.
• Time 2 weeks
77