YIELD DESIGN ANALYSIS

In a general case, the yield design is used to determine the ultimate load that a structure can sustain. Two approaches can be used: an interior (static) approach, which is based on statically admissible stress fields and gives the lower bound of the ultimate load; or an exterior (kinematic) approach, which is based on the virtual work theorem with the study of kinemati-cally admissible virtual velocity fields and gives the upper bound of the ultimate load. Colas et al. (2008, 2010a,b) chose a kinematic approach

in combination with the homogenisation theory developed for periodic masonry (de Buhan and de Felice 1997) to model drystone retaining walls.

Yield design theory requires three kinds of parameters: geometry of the system, loading mode and resistance of the constituent material. In the stud-ied case, the problem geometry is defined by a height H, thickness at the bottom B, front batter α and backfill height Hb (Figure 4.4). As would be expected, there is a strong degree of similarity between Figures 4.2 and 4.4.

The loadings considered in the study are the respective unit weights, γDW

and γs, of the wall and its backfill soil. The wall was approximated as built from rigid regular cut stone blocks with dimensions a and b, so that it could be considered as periodic. The joints are assumed to have a purely frictional Mohr–Coulomb shear criterion, depending only on the block friction angle φb.

It is then possible to consider the cell represented in Figure 4.5 to imple-ment the homogenisation process in the framework of yield design theory.

Figure 4.4 Drystone retaining wall modelling.

K

J I

h

O O´

B β

ψ f₁

γ γ_s

ψ_s

1 2

a

b 1

2

b

a

(a) (b)

Figure 4.5 Cell of the periodic masonry. (a) Actual wall. (b) Unit cell.

The macroscopic strength domain G^hom, describing the set of macro-scopic stress states Σ such that there exists a stress field σ defined over the cell (C) and verifying the following conditions, is written

Σ =^σ

( )

^{x =V1}cell

∫

_C^σ

( )

^{x dV} (4.4)

div σ(x) = 0 (4.5)

with σ ⋅n x antiperiodic, with n(x) being the unit normal oriented out-

( )

wards from the cell C, and σ(x) ∈ G(x) whatever x ∈ C, G(x) characterising the strength capacities of the constituent materials.

In this approach, the kinematic definition of G^hom will be used, which can be obtained through the dualisation of the static definition by means of the principle of virtual work. One considers any virtual velocity field of the form

v x

( )

⁼Fx u x⁺

( )

(4.6)

with F any second-order tensor and u a periodic velocity field. The strain rate field d can be written as

d x

( )

^{= +}D ^δ

( )

x (4.7)

where D is the symmetric part of F and δ is the strain rate field associ-ated with u.

The principle of virtual work leads to:

σ: d dV σ n v dS Vσ Vv

C C

∫

⁼_∂

∫

^{⋅ ⋅} Statically Admissible, (4.8)

By replacing v by its expression, and because u is periodic and σ ⋅n is anti-periodic, it follows that

σ:d dV σdV F:

C C

∫

⁼

∫

^(4.9)

so

σ:d =Σ:F=Σ:D (4.10)

where

. =^V1cell

∫

_C⋅^dV

(4.11)

Introducing

π σ σ π

d σ d G x D D G

( )

^=max

{

^{: , ∈}

^{( )} }

^{and hom}

( )

^=max_Σ

{

^{Σ: ,Σ∈ homm}

}

^(4.12)

which are the respective support functions of the convex domain G(x) and G^hom. The static definition of G^hom leads to

π^hom

( )

D ^π

( )

d Vuperiodic ^(4.13)

Then it may be proven that π^hom D π d

( )

^=minu

( )

^(4.14)

which constitutes the kinematic definition of G^hom. We may then consider a particular velocity field v defined as

v(x) = vⁱ (4.15)

where vⁱ is the velocity of any block Cⁱ (see Figure 4.6).

The obtained strength domain is represented in Figure 4.7 in the field (Σ11, Σ22, Σ12). m is the slenderness ratio (a/b) of the blocks and f = tan φb, with φb being the friction angle between the blocks.

The support function of the strength domain (Figure 4.7) of the homogenised material is obtained as

π^hom(n,(v)) = 0 (4.16)

Figure 4.6 Velocity field solution in the cell.

1 2

b

a C₃

C₂

C1

C₄ v₂

v₁

with the conditions

− ≤

( )

^≤

≤ −

( )

n v

n v mn v

n v n v n v

1 1

1 1 2 2

1 2 2 1 1 1

0 2 tan

+ tan

b

b

φ

φ ++n v_{2 2}/tan

( )

φb ^(4.17)

else

π^hom(n,(v)) = +∞ (4.18)

where v is the discontinuity of the virtual velocity field in the wall and v_i (i = 1, 2) are the components of the velocity v of the cell constitutive blocks.

The soil is considered to be a Mohr–Coulomb material, depending on its cohesion C_s and friction angle φs characterised by

π d_s C^sφ d d d d φ

s s s s s s

tan tr if tr sin

( )

⁼

_{( )}

      ^≥

(

^{1+ 2}

) ^{( )}

^(4.19)

Σ₁₂ Σ₂₂

Σ₁₁

(−1,0,0)

(−f,1,−1/f ) (−f,−1,−1/f )

(−f,0,−2 m)

Figure 4.7 Homogenised strength domain of drystone retaining wall.

where d_sis the rate deformation tensor and π n v_{s s} Cφ v n v n v φ

s

s s s s s s

tan if sin

(

,

)

⁼

( )

^{⋅ ⋅ ≥}

( )

(4.20)

with v_s any discontinuity of the velocity field in the soil.

The interface strength criterion between the back face of the wall and the backfill is described by a frictional Coulomb interface:

π(n, Δv) = 0 if n·Δv ≥ |Δv|sin(δ) (4.21) where Δv represents the velocity discontinuity and δ the friction angle between the soil and the wall. This friction angle will be taken as

δ = min {ϕs, ϕb} (4.22)

This choice is justified by the fact that the optimal discontinuity line will always be localised in the medium with the smaller friction angle.

Second, the yield design was applied to calculate an estimation of the ultimate backfill height.

Noting that the back of the wall is not smooth but quite rough, the fric-tion angle at the wall-backfill interface δ was set equal to the backfill fric-tion angle φs. Knowing all necessary parameters, the possible ultimate backfill height was calculated in the framework of the kinematic approach of the yield design theory (Salençon 1983, 2013). This theory is based on the principle of virtual work combined with the knowledge of the strength criterion and leads to the inequality between the work of external actions (W^e) and the maximum resisting work (W^mr) for all kinematically admis-sible virtual velocity fields as a necessary condition of stability:

W^eW^mr∀   V KA (4.23)

Two different mechanisms of failure will be shown here: translation of the wall and soil (Figure 4.8a) and wall rotation and soil shearing (Figure

(a) (b)

Figure 4.8 Studied failure mechanisms. (a) Translation of the wall and soil and (b) wall rotation and soil shearing.

4.8b). The smaller result of the two cases was taken as the final result.

These last failure mechanisms were verified by 2D scale-down tests using Schneebeli rods to simulate backfill soil in two dimensions (Colas et al.

2010a).

It should be noted that the backfill height is considered different from the height of the wall, and the wall stability is evaluated in relation to the soil height that the wall could support. The stability problem could be charac-terised by the following nondimensional factor depending on nondimen-sional parameters:

The factor of safety in this case can be defined as

F H

= H^breal

blimit (4.25)

in which H_b^real is the actual height of the backfill soil, while H_b^limit is the maximum height of the backfill soil required to ensure stability.

Although the yield design theory is more complicated than the limit equi-librium, it has been considered to give better results than the approach of Villemus, which depended on the value of the angle ψ between the hori-zontal and the failure line through the wall, which can be estimated by measurement, whereas in the yield design this angle is calculated in the optimisation process. However, this process depends on the geometry of the construction in the same way as in the limit equilibrium approach of Villemus, because that failure plane steps up through the courses of masonry in exactly the same way as it can in the homogenisation. The difference is that homogenisation implicitly allows steeper angles provided that they also step through the structure in a similar way, whereas Villemus implicitly checked for just ψ = 0 and for the first stepping value. A thorough limit equilibrium check would, as a matter of course, consider these mecha-nisms also, but observations of test walls confirm the assumption that those considered by Villemus would normally be critical. In all cases, a conscious decision must be made to consider steeper values of ψ, and the actual pos-sible values depend on the geometry of the stone used in the construction.

In document (Applied Geotechnics) Paul F. McCombie, Jean-Claude Morel, Denis Garnier-Drystone Retaining Walls_ Design, Construction and Assessment-CRC Press (2015) (Page 92-98)