The continuum and its coherence Stability analysis of a retaining wall

The continuum and its coherence

Stability analysisof a retaining wall






Elementary surface forces




Augustin CAUCHY1789 -1857

Exercices de mathématiques (1829)

S

S

M

S

M

M

M

zyzx

xx

xy

xz

x

y

zzz

yyyz

yx

M

zyzx

xx

xy

xz

x

y

zzz

yyyz

yx

x

y

z

M

x

y

z anTf d)(d

ad

n

x

y

z anTf d)(d

ad

n

jjii nT

Linearity

x

y

z

M

jjii nT

Linearity

M

zyzx

xx

xy

xz

x

y

zzz

yyyz

yxjjii nT

Linearity

M

zyzx

xx

xy

xz

x

y

zzz

yyyz

yxjjii nT

Linearity

M

zyzx

xx

xy

xz

x

y

zzz

yyyz

yxjjii nT

jiij

0)(

iij

ji aFx

Linearity

Symmetry

Equations of dynamics

jjii nT

jiij

0)(

iij

ji aFx

Linearity

Symmetry


Cauchy stress TENSOR

Classical presentationof the

for modelling

INTERNAL FORCES

jjii nT

jiij

0)(

iij

ji aFx

Linearity

Symmetry


Cauchy stress TENSOR

Classical presentationof the

does not refer to any Stability

or Rupture analysis

Potential collapse mechanisms

Rotation about B

Rigid body motion

Physical feeling of the Mathematical duality between internal forces and deformation of matter

Physical feeling of the Mathematical duality between internal forces and deformation of matter

The virtual work methodThe virtual work method

GeometricalModel


PRINCIPLEofvirtual work

Appropriate choice of virtual motions

GeometricalModel


DUALITY



GeometricalModel


DUALITY


GeometricalModel

Representationof FORCES



DUALITY

Dimensional analysis


Yield design



Yield design

Yield design Galileo

Yield design

P

hb

0B

l

Galileo

Yield design

P

hb

0B

l

considers that the beam acts as a lever with fulcrum in B.

Galileo

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

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


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


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


P

hb

0B

l

Galileo

Coulomb

for the loads that can be sustainedby the system under these conditions




for the loads that can be sustainedby the system under these conditions



K

jQ

iQO



Resistance of the materialsare mathematically compatible

K

jQ

iQO

The domain of potentially safe loads

is convex




K


is convex

jQ

iQO

Interior estimate




K


is convex

jQ

iQO

Exterior estimate?

must be mathematically compatible



Resistance of the materials K

jQ

iQO

a CONVEX domainis assigned to the STRESS stateat any point of the system




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




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


K

jQ

iQO


• Constructing virtual velocity fields


K

jQ

iQO


• Constructing virtual velocity fields• Writing the balance


K

jQ

iQO• Constructing virtual velocity fields• Writing the balance

betweenthe external forces rate of work



K

jQ


betweenthe external forces rate of work andthe maximum resisting rate of work



K

jQ




Exterior estimate


K

jQ




Exterior estimate

Support function defined by duality

P

hb

0B

l

Galileo

lPhb

2

2

0


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


Coulomb

The virtual collapse mechanism is a rigid body motion of BegC.

Exterior estimate of the stability of the wall


Ultimate Limit State 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












N.K. OVESEN












N.K. OVESEN












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.



Yield design



Yield design

Documents

The continuum and its coherence Stability analysis of a retaining wall