The continuum and its coherence Stability analysis of a retaining wall

The continuum and its coherence Stability analysis of a retaining wall

The continuum and its coherence Stability analysis of a retaining wall

The continuum and its coherence Elementary surface forces Stability analysis of a retaining wall

The continuum and its coherence

Augustin CAUCHY Exercices de mathématiques (1829)

Linearity

Symmetry Equations of dynamics

Linearity Symmetry Equations of dynamics Cauchy stress TENSOR Classical presentation of the for modelling INTERNAL FORCES

Linearity Symmetry Equations of dynamics Cauchy stress TENSOR Classical presentation of 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

The virtual work method

Geometrical Model The virtual work method

PRINCIPLE of virtual work Appropriate choice of virtual motions Geometrical Model The virtual work method DUALITY

PRINCIPLE of virtual work Appropriate choice of virtual motions Geometrical Model The virtual work method DUALITY

PRINCIPLE of virtual work Geometrical Model Representation of FORCES Appropriate choice of virtual motions The virtual work method DUALITY

Dimensional analysis The continuum and its coherence Yield design

Dimensional analysis The continuum and its coherence Yield design

Galileo

Yield design Galileo

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

Yield design resistance for the wood fibers on the other hand, assuming that they are in their limit state of tension. writes the balance equation for the moments at point B for the active load on the one hand Galileo

Yield design Coulomb

Yield design Coulomb resistance and the resistance of the material along Beg writes the balance equation between the active forces should look for the most unfavourable partition

The Theory of Yield design Galileo Coulomb Geometry of the system Multi-parameter loading process Resistance of the constituent materials a CONVEX domain is assigned to the STRESS state at any point of the system

The Theory of Yield design Galileo Coulomb Geometry of the system Multi-parameter loading process Resistance of the constituent materials What loads can be sustained by the system under these conditions

The Theory of Yield design Galileo Coulomb What loads can be sustained by the system under these conditions Equilibrium of the system Resistance of the materials must be mathematically compatible

The Theory of Yield design Galileo Coulomb for the loads that can be sustained by the system under these conditions Equilibrium of the system Resistance of the materials must be mathematically compatible

The Theory of Yield design for the loads that can be sustained by the system under these conditions Equilibrium of the system Resistance of the materials must be mathematically compatible

The Theory of Yield design Equilibrium of the system Resistance of the materials are mathematically compatible The domain of potentially safe loads is convex

The Theory of Yield design Equilibrium of the system Resistance of the materials are mathematically compatible The domain of potentially safe loads is convex Interior estimate

The Theory of Yield design Equilibrium of the system Resistance of the materials are mathematically compatible The domain of potentially safe loads is convex Exterior estimate?

must be mathematically compatible The Theory of Yield design Equilibrium of the system Resistance of the materials a CONVEX domain is assigned to the STRESS state at any point of the system

The Theory of Yield design Equilibrium of the system Resistance of the materials must be mathematically compatible a CONVEX domain is assigned to the STRESS state at any point of the system the CONVEX domain is defined by DUALITY at any point of the system through its SUPPORT FUNCTION on the VIRTUAL STRAIN RATES

The Theory of Yield design Equilibrium of the system Resistance of the materials must be mathematically compatible DUAL DEFINITION of the convex domain of potentially safe loads the CONVEX domain is defined by DUALITY at any point of the system through its SUPPORT FUNCTION on the VIRTUAL STRAIN RATES

The Theory of Yield design DUAL DEFINITION of the convex domain of potentially safe loads Constructing virtual velocity fields

The Theory of Yield design DUAL DEFINITION of the convex domain of potentially safe loads Constructing virtual velocity fields Writing the balance

The Theory of Yield design Constructing virtual velocity fields Writing the balance between the external forces rate of work DUAL DEFINITION of the convex domain of potentially safe loads

The Theory of Yield design Constructing virtual velocity fields Writing the balance between the external forces rate of work and the maximum resisting rate of work DUAL DEFINITION of the convex domain of potentially safe loads

The Theory of Yield design Constructing virtual velocity fields Writing the balance between the external forces rate of work and the maximum resisting rate of work DUAL DEFINITION of the convex domain of potentially safe loads Exterior estimate

The Theory of Yield design Constructing virtual velocity fields Writing the balance between the external forces rate of work and the maximum resisting rate of work DUAL DEFINITION of the convex domain of potentially safe loads Exterior estimate Support function defined by duality

Galileo The Theory of Yield design The virtual collapse mechanism is a rotation about fulcrum B.

Galileo 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

Ultimate Limit State Design 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: R d ≥ S d S d is the design load effect calculated on the basis of the principles … The design resistance effect R d 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: R d ≥ S d S d is the design load effect calculated on the basis of the principles … The design resistance effect R d 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: R d ≥ S d S d is the design load effect calculated on the basis of the principles … The design resistance effect R d 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: R d ≥ S d S d is the design load effect calculated on the basis of the principles … The design resistance effect R d which in the case of the design of a footing is the design ultimate bearing capacity … N.K. OVESEN

For practical implementation to the design of structures this symbolical inequality must be given a quantitative significance Design RESISTANCE Effect Design LOAD Effect

For practical implementation to the design of structures this symbolical inequality is given a quantitative significance Design RESISTANCE Effect Design LOAD Effect through the dual approach within the theory of yield design.

Dimensional analysis The continuum and its coherence Yield design

Dimensional analysis The continuum and its coherence Yield design