eciv 520 a structural analysis ii lecture 4 – basic relationships
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![Page 1: ECIV 520 A Structural Analysis II Lecture 4 – Basic Relationships](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d355503460f94a0cae2/html5/thumbnails/1.jpg)
ECIV 520 A Structural Analysis II
Lecture 4 – Basic Relationships
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Process of Matrix/FEM Analysis
Accurate Approximations of Solutions
Reliability of Solution depends on choice of Mathematical Model
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Process of Matrix/FEM Analysis
Reliability of Solution depends on choice of Mathematical Model
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Process of Matrix/FEM Analysis
Accurate Approximations of Solutions
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Process of Matrix/FEM Analysis
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Process of Matrix/FEM Analysis
Accurate Approximations of Solutions
Reliability of Solution depends on choice of Mathematical Model
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StressResultant Force and Moment represent the resultant effects of the actual distribution of force acting over sectioned area
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Stress
AssumptionsMaterial is continuousMaterial is cohesive
Force can be replaced by the three componentsFx, Fy (tangent) Fz (normal)
Quotient of force and area is constantIndication of intensity of force
Consider a finite but very small area
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Normal & Shear Stress
Normal StressIntensity of force acting normal to A
Shear StressIntensity of force acting tangent to A
dA
dF
A
F zz
Az
0
lim
dA
dF
A
F xx
Azx
0
limdA
dF
A
F yy
Azy
0
lim
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General State of Stress
Set of stress components depend on orientation of cube
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Basic Relationships of Elasticity Theory
Concentrated
Distributed on Surface
Distributed in Volume
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Equilibrium
Equilibrium
Write Equations of Equilibrium
Fx=0
Fy=0
Fz=0
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EquilibriumEquilibrium - X
X
dAdzz
dA
dAdzz
dA
dAdxx
dA
xzxz
xz
xyxy
xy
xx
x
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EquilibriumEquilibrium
Write Equations of Equilibrium
Fx=0
Fy=0
Fz=0
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Boundary Conditions
Prescribed Displacements
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Boundary Conditions
Equilibrium at Surface
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Deformation
Intensity of Internal Loads is specified using the concept of
Normal and Shear STRESS
Forces applied on bodies tend to change the body’s
SHAPE and SIZE
Body Deforms
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Deformation
• Deformation of body is not uniform throughout volume
• To study deformational changes in a uniform manner consider very short line segments within the body (almost straight)
Deformation is described by changes in length of short line segments and the
changes in angles between them
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Strain
Deformation is specified using the concept of
Normal and Shear STRAIN
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Normal Strain - Definition
Normal Strain: Elongation or Contraction of a line segment per unit of length
s
ss
'
avgε
s
ssAB
'
n along limε
ss ε1'
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Normal Strain - Units
Dimensionless Quantity: Ratio of Length Units
Common Practice
SI
m/mm/m (micrometer/meter)
US
in/in
Experimental Work: Percent
0.001 m/m = 0.1%
=480x10-6: 480x10-6 in/in 480 m/m 480 (micros)
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Shear Strain – DefinitionShear Strain: Change in angle that occurs between two line
segments that were originally perpendicular to one another
(rad) 'lim2
talongACnalongAB
nt
negative 2
'
positive 2
'
nt
nt
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Cartesian Strain Components
zz
yy
xx
ε1
ε1
ε1
zx
yz
xy
2
2
2
Normal Strains: Change VolumeShear Strains: Change Size
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Small Strain Analysis
Most engineering design involves application for which only small deformations are allowed
DO NOT CONFUSE Small Deformations with Small Deflections
Small Deformations => <<1
Small Strain Analysis: First order approximations are made about size
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Strain-Displacement Relations
AssumptionSmall Deformations
For each face of the cube
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Stress-Strain (Constitutive)Relations
Isotropic Material: E,
Generalized Hooke’s Law
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Stress-Strain (Constitutive)Relations
Note that:
Equations (a) can be solved for
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Stress-Strain (Constitutive)Relations
Or in matrix form
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Stress-Strain (Constitutive)Relations
xy
xz
yz
z
y
x
xy
xz
yz
z
y
x
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Stress-Strain: Material Matrix
Material Matrix
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Special Cases
One Dimensional: v=0No Poisson Effect
= E
Reduces to:
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Special CasesTwo Dimensional – Plane StressThin Planar Bodies subjected to in plane loading
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Special CasesTwo Dimensional – Plane Strain
Long Bodies Uniform Cross Section subjected to transverse loading
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Special Cases
For other situations such as inostropy obtain the appropriate material matrix
Two Dimensional – Plane Stress Orthotropic Material
Dm=
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Strain EnergyDuring material deformation energy is stored (strain energy)
yxAF zL
V
zyx
zFU
2
1
2
1
2
1
Strain Energy Density
2
1
V
Uu
e.g. Normal Stress
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Strain Energy
In the general state of stress for conservative systems
dVUV
T εσ2
1
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Principle of Virtual Work
PFdxWPx
2
1
0
Load Applied Gradually Due to another Force
'PWP
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Apply Real LoadsApply Virtual Load
PVW Concept
Real
Deformns
udLP '
Internal
Virtual
Forces
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Principle of Virtual Work
A body is in equilibrium if the internal virtual work equals the external virtual work for every
kinematically admissible displacement field
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Principle of Virtual Work
V
zx
yz
xy
z
y
x
zxyzxyzyx
V
Tie
dV
dVUW
εσ