strain and deformation γfast10.vsb.cz/lausova/lesson01.pdf · 5 / 33 geometrical equations x y ma...
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Department of Structural Mechanics
Faculty of Civil Engineering, VSB - Technical University Ostrava
Elasticity and Plasticity
Strain and Deformation
• Deformations and Displacements of 3D solid
• Physical Relations between Strains and Deformations• Hook’s Law, Physical Constants and Stress-Strain
Diagram of Construction Materials• Deformation by the Temperature Changes
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Deformations and displacements
Deformations and Displacements of 3D solid
3D bodies are deformated by influence of loading or
temperature changes,it is possible to define relative deformations or displacement components.
Relative deformations:
- length ε (relative elongation or contraction)
- angular γ (cross-section tapering)
Small-deformation theory: 1
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Geometrical equations
x
y
M A
B C
C’
B’
A’
M’
dy
dx
dx’
v
u
x
u
x
xuxx
uux
x
xxx
∂
∂=
−
−
∂
∂++
=−′
=d
ddd
d
ddε
xx
uu d
∂
∂+
Deformations and Displacements of 3D solid 6 / 33
Geometrical equations
x
y
M A
B C
C’
B’
A’
M’
dy
dx
dx’
dy’
v
u
xx
uu d
∂
∂+
xx
vd
∂
∂
yy
ud
∂
∂
β
α
y
u
x
v
y
yy
u
x
xx
v
xy∂
∂+
∂
∂=
∂
∂
+∂
∂
=+=d
d
d
d
βαγ
Deformations and Displacements of 3D solid
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Geometrical equations
x
ux
∂
∂=ε
y
vy
∂
∂=ε
z
wz
∂
∂=ε
y
u
x
vxy
∂
∂+
∂
∂=γ
z
v
y
wyz
∂
∂+
∂
∂=γ
x
w
z
uzx
∂
∂+
∂
∂=γ
Deformations and Displacements of 3D solid
Relations between components of relative deformationin 3D body and components of displacements of any points of 3D body are expressed by geometrical equations.
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Stress-strain diagram of construction materials
Physical Relations between Strains and Deformations
ε
σ
Relations between strains and deformations are expressed by stress-strain diagram. It depends on physical and mechanical properties of construction materials.
Tension
A
N
Ar
r
∆
∆=
→∆ 0limσ
A
N=σ
l
l∆=ε
Dir
ect
str
ess
Relative deformation
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N
N
Physical Relations between Strains and Deformations
Tensile stress
Tensile test of steel
Stress-strain diagram of construction materials
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N
N
Physical Relations between Strains and Deformations
Tensile stress
Tensile test of steel
Stress-strain diagram of construction materials
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N
N
Physical Relations between Strains and Deformations
Tensile stress
Tensile test of steel
Stress-strain diagram of construction materials
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Broken steel
specimen after tensile test
Physical Relations between Strains and Deformations
Tensile stress
Stress-strain diagram of construction materials
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Tensile stress
Physical Relations between Strains and Deformations
Broken steel
specimen after tensile test
Stress-strain diagram of construction materials
14 / 33Physical Relations between Strains and Deformations
Tensile test of steel,
stress-strain diagram
Stress-strain diagram of construction materials
15 / 33Physical Relations between Strains and Deformations
Tensile test of steel,
stress-strain diagram
Stress-strain diagram of construction materials
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Tensile test of steel, stress-strain diagram
Physical Relations between Strains and Deformations
Stress-strain diagram of construction materials
-
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Tensile test of steel, stress-strain diagram
ε
σ
Dir
ect
str
ess
Relative deformation
Linear-elastic matherial
Physical Relations between Strains and Deformations
Stress-strain diagram of construction materials
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ε
σ
Plastic behaviour of matherial
Physical Relations between Strains and Deformations
Tensile test of steel, stress-strain diagram
Dir
ect
str
ess
Relative deformation
Stress-strain diagram of construction materials
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ε
σ
Permanent deformation
ach
ievem
ent in
tensile
test
Physical Relations between Strains and Deformations
Tensile test of steel, stress-strain diagram
Dir
ect
str
ess
Relative deformation
Stress-strain diagram of construction materials
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Linear-elastic matherial, Hook’s law
εx
Tension
EE xx
x
x σεε
σα =→== tan
A
Nx =σ
l
lx
∆=ε
α = arctan E
E
A
N
l
l=
∆
σx ... direct stress [Pa]
εx ... relative deformation [-]
AE
lNl
.
.=∆→
Hook’s law
σ
E ... modulus of elasticity in tension andcompression (Young’s modulus) [Pa]
Physical Relations between Strains and Deformations
Dir
ect
str
ess
Relative deformation
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Linear-elastic matherial, Hook’s law
υ ... Poisson's ratio [-]
dx
σx σx
dy
dx’
dy’
after deformation
E
xxzy
συευεε .. −=−==
5,0≤υ
In simultaneously operation
of σx, σy and σz
( )[ ]zyxzyxxEEEE
σσυσσ
υσ
υσ
ε +−=−−= ..1
..
likewise ( )[ ]zxyyE
σσυσε +−= ..1 ( )[ ]
yxzzE
σσυσε +−= ..1
Physical equation – 1st part
Physical Relations between Strains and Deformations 22 / 33
was a British natural philosopher, architect and polymath who played an important role in the Scientific Revolution, through both experimental and theoretical work. In 1660, Hooke discovered the law of elasticity which bears his name and which describes the linear variation of tension with extension in an elastic spring. He first described this discovery in the anagram "ceiiinosssttuv", whose solution he published in 1678 as "Ut tensio, sic vis" meaning "As the extension, so the force.”
Historical persons
Robert Hooke
Thomas Young
Siméon-Denis Poisson
was an English genius and polymath. He is famous with the public for having partly deciphered Egyptian hieroglyphs before Champollion did. Young made notable scientific contributions to the fields of vision, light, solid mechanics, energy, physiology, language, musical harmony and Egyptology. Young described the characterization of elasticity that came to be known as Young's modulus, denoted as E, in 1807, and further described it in his subsequent works such as his 1845 Course
of Lectures on Natural Philosophy and the Mechanical Arts.
was a French mathematician, geometer, and physicist. After him is named
Poisson's ratio, when a sample object is stretched, of the contraction or transverse strain (perpendicular to the applied load), to the extension or axial strain (in the direction of the applied load).
Physical Relations between Strains and Deformations
(18 July 1635 – 3 March 1703)
(13 June 1773 – 10 May 1829)
(21 June 1781 – 25 April 1840)
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Shear, shear stresses
x
y
dy
dx
βafter d
eformation
τxyτxy
τyx
τyx
Physical Relations between Strains and Deformations 24 / 33
Linear-elastic matherial, Hook’s law in shear
γxy
τxy = τyx
α = arctan G
τxy ... shear stress [Pa]
γxy ... tapering cross-section [-]
G ... modulus of elasticity in shear [Pa]
Hook’s law in shear
GG
xy
xy
xy
xy τγγ
τα =→== tan
likewise
G
yz
yz
τγ =
G
zxzx
τγ =
Physical equation – 2nd part
Physical Relations between Strains and Deformations
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Physical equations
Physical equations expressing relations between components of relative deformations and components of strains in 3D body.
G
yz
yz
τγ =
G
zxzx
τγ =
G
xy
xy
τγ =
( )[ ]zxyy
Eσσυσε +−= ..
1
( )[ ]yxzz
Eσσυσε +−= ..
1
( )[ ]zyxx
Eσσυσε +−= ..
1
Physical Relations between Strains and Deformations 26 / 33
Physical constants
In case of isotropic substance is not E, G and υ mutual indipendence.
5,00 ≤≤ υ( )υ+= 1.2G
E→
23
EG
E≤≤
Benchmark values of physical constants of some matherials:
E G υ
Steel 210 000 MPa 81 000 MPa 0,3
Glass 70 000 MPa 28 000 MPa 0,25
Granite 12 000 to 50 000 MPa - 0,2
SoftwoodE|| = 10 000 MPa
E⊥ = 300 MPa600 MPa -
Physical Relations between Strains and Deformations
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Stress-strain diagram of concrete in compression
Physical Relations between Strains and Deformations 28 / 33
Physical constants of concrete
Grade of concreteEcm
modulus of elasticityG υ
C12 26 000 MPa
0,42.E 0,2
C16 27 500 MPa
C20/25 29 000 MPa
C25/30 30 500 MPa
C30/35 32 000 MPa
C35/45 33 500 MPa
C40/50 35 000 MPa
C45/55 36 000 MPa
C50/60 37 000 MPa
Physical Relations between Strains and Deformations
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Design stress-strain diagram of concrete in compression
Parabolic-rectangular
Idealized diagram
Design diagram
Physical Relations between Strains and Deformations 30 / 33
Bilinear
Idealized diagram
Design diagram
Physical Relations between Strains and Deformations
Design stress-strain diagram of concrete in compression
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Stress-strain diagram of steel
Plasticity: matherial ability to permanent deformation without fracture.
fe … limit of elasticity
fy ... yield stressfu ... ultimate strength
Steel ductility: plastic elongation of broken bar, steel 15%.
Strain energy
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Ideal elasto-plastic matherial
εx
σx
Compression
α = arctan E
Tension
fy
0
Y
Y’
A,C
B
section
Y-Y’ Hook’s law
Y-A Plastic state – free increase of
deformations
A-B Unloading
B-C Re-increasing of strain
εp εe
εp … plastic (permanent) deformationεe … elastic deformation
-fy
Physical Relations between Strains and Deformations
Stress-strain diagram
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Deformation by Temperature Changes
Deformation by Temperature Changes
dy
dx
( )CT o∆
dx’
dy’
TTTzTyTx ∆=== .,,, αεεε 0=== zxyzxy γγγ
αt … coefficient of thermal expansivity [oC-1]
Steel αt=12.10-6 oC-1 Timber α
t=3.10-6 oC-1
Concrete αt=10.10-6 oC-1 Masonry α
t=5.10-6 oC-1