The continuum and its coherence Stability analysis of a retaining wall.
-
Upload
charleen-francis -
Category
Documents
-
view
215 -
download
2
Transcript of The continuum and its coherence Stability analysis of a retaining wall.
jjii nT
jiij
0)(
iij
ji aFx
Linearity
Symmetry
Equations of dynamics
Cauchy stress TENSOR
Classical presentationof the
for modelling
INTERNAL FORCES
jjii nT
jiij
0)(
iij
ji aFx
Linearity
Symmetry
Equations of dynamics
Cauchy stress TENSOR
Classical presentationof the
does not refer to any Stability
or Rupture analysis
Physical feeling of the Mathematical duality between internal forces and deformation of matter
The virtual work methodThe virtual work method
PRINCIPLEofvirtual work
Appropriate choice of virtual motions
GeometricalModel
The virtual work methodThe virtual work method
DUALITY
PRINCIPLEofvirtual work
Appropriate choice of virtual motions
GeometricalModel
The virtual work methodThe virtual work method
DUALITY
PRINCIPLEofvirtual work
GeometricalModel
Representationof FORCES
Appropriate choice of virtual motions
The virtual work methodThe virtual work method
DUALITY
Yield design
P
hb
0B
l
resistance
• for the wood fibers on the other hand, assuming that they are in their limit state of tension.
writes the balance equationfor the moments at point B• for the active load on the one hand
Galileo
lPhb
2
2
0
Yield design Coulomb
resistance
• and the resistance of the material along Beg
writes the balance equationbetween• the active forces
• should look forthe most unfavourable partition
The Theory of Yield design
P
hb
0B
l
Galileo
Coulomb
• Geometry of the system• Multi-parameter loading process• Resistance of the constituent materials
a CONVEX domainis assigned
to the STRESS stateat any point of the system
The Theory of Yield design
P
hb
0B
l
Galileo
Coulomb
• Geometry of the system• Multi-parameter loading process• Resistance of the constituent materials
What loads can be sustainedby the system under these conditions
The Theory of Yield design
P
hb
0B
l
Galileo
Coulomb
What loads can be sustainedby the system under these conditions
Equilibrium of the system
Resistance of the materialsmust be mathematically compatible
The Theory of Yield design
P
hb
0B
l
Galileo
Coulomb
for the loads that can be sustainedby the system under these conditions
Equilibrium of the system
Resistance of the materialsmust be mathematically compatible
The Theory of Yield design
for the loads that can be sustainedby the system under these conditions
Equilibrium of the system
Resistance of the materialsmust be mathematically compatible
K
jQ
iQO
The Theory of Yield design
Equilibrium of the system
Resistance of the materialsare mathematically compatible
K
jQ
iQO
The domain of potentially safe loads
is convex
The Theory of Yield design
Equilibrium of the system
Resistance of the materialsare mathematically compatible
K
The domain of potentially safe loads
is convex
jQ
iQO
Interior estimate
The Theory of Yield design
Equilibrium of the system
Resistance of the materialsare mathematically compatible
K
The domain of potentially safe loads
is convex
jQ
iQO
Exterior estimate?
must be mathematically compatible
The Theory of Yield design
Equilibrium of the system
Resistance of the materials K
jQ
iQO
a CONVEX domainis assigned to the STRESS stateat any point of the system
The Theory of Yield design
Equilibrium of the system
Resistance of the materialsmust be mathematically compatible
K
jQ
iQO
a CONVEX domainis assigned to the STRESS stateat any point of the system
the CONVEX domainis defined by DUALITYat any point of the systemthrough its SUPPORT FUNCTIONon the VIRTUAL STRAIN RATES
The Theory of Yield design
Equilibrium of the system
Resistance of the materialsmust be mathematically compatible
K
jQ
iQO
DUAL DEFINITION ofthe convex domain of potentially safe loads
the CONVEX domainis defined by DUALITYat any point of the systemthrough its SUPPORT FUNCTIONon the VIRTUAL STRAIN RATES
The Theory of Yield design
K
jQ
iQO
DUAL DEFINITION ofthe convex domain of potentially safe loads
• Constructing virtual velocity fields
The Theory of Yield design
K
jQ
iQO
DUAL DEFINITION ofthe convex domain of potentially safe loads
• Constructing virtual velocity fields• Writing the balance
The Theory of Yield design
K
jQ
iQO• Constructing virtual velocity fields• Writing the balance
betweenthe external forces rate of work
DUAL DEFINITION ofthe convex domain of potentially safe loads
The Theory of Yield design
K
jQ
iQO• Constructing virtual velocity fields• Writing the balance
betweenthe external forces rate of work andthe maximum resisting rate of work
DUAL DEFINITION ofthe convex domain of potentially safe loads
The Theory of Yield design
K
jQ
iQO• Constructing virtual velocity fields• Writing the balance
betweenthe external forces rate of work andthe maximum resisting rate of work
DUAL DEFINITION ofthe convex domain of potentially safe loads
Exterior estimate
The Theory of Yield design
K
jQ
iQO• Constructing virtual velocity fields• Writing the balance
betweenthe external forces rate of work andthe maximum resisting rate of work
DUAL DEFINITION ofthe convex domain of potentially safe loads
Exterior estimate
Support function defined by duality
P
hb
0B
l
Galileo
lPhb
2
2
0
The Theory of Yield design
The virtual collapse mechanism is a rotation about fulcrum B.
P
hb
0B
l
The virtual collapse mechanism is a rotation about fulcrum B.
Galileo
lPhb
2
2
0 Exterior estimate
The Theory of Yield design
Coulomb
The virtual collapse mechanism is a rigid body motion of BegC.
Exterior estimate of the stability of the wall
The Theory of Yield design
According to the principle of Limit
States Design, the design criterion is
simply to design for equilibrium in the
design limit state of failure. The design
criterion could be expressed in the
following way:Rd ≥ Sd
Sd is the design load effect calculated
on the basis of the principles …
The design resistance effect Rd which
in the case of the design of a footing is
the design ultimate bearing capacity …
N.K. OVESEN
According to the principle of Limit
States Design, the design criterion is
simply to design for equilibrium in the
design limit state of failure. The design
criterion could be expressed in the
following way:Rd ≥ Sd
Sd is the design load effect calculated
on the basis of the principles …
The design resistance effect Rd which
in the case of the design of a footing is
the design ultimate bearing capacity …
N.K. OVESEN
According to the principle of Limit
States Design, the design criterion is
simply to design for equilibrium in the
design limit state of failure. The design
criterion could be expressed in the
following way:Rd ≥ Sd
Sd is the design load effect calculated
on the basis of the principles …
The design resistance effect Rd which
in the case of the design of a footing is
the design ultimate bearing capacity …
N.K. OVESEN
According to the principle of Limit
States Design, the design criterion is
simply to design for equilibrium in the
design limit state of failure. The design
criterion could be expressed in the
following way:Rd ≥ Sd
Sd is the design load effect calculated
on the basis of the principles …
The design resistance effect Rd which
in the case of the design of a footing is
the design ultimate bearing capacity …
N.K. OVESEN
dd SR
dd SR
For practical implementation to the design of structures this symbolical inequalitymust be givena quantitative significance
DesignRESISTANCEEffect
DesignLOADEffect
dd SR
For practical implementation to the design of structures this symbolical inequalityis givena quantitative significance
DesignRESISTANCEEffect
DesignLOADEffect
through the dual approach within the theory of yield design.