3.5.1 Problem definition
We consider the pavement as a rectangular parallelepiped that is subjected to a surface contact loading limited in a circle of radius a in point contact or in a rectangle of a × b in line contact, as shown in figures (3.3) and (3.4). The pavements are subjected time-dependent loads p(t) = µ1(t)p0 and q(t) = µ2(t)q0
where µ1, µ2 ≥ 0, each of them can vary independently within a given range.
These loads form a convex polyhedral domain L of nl dimensions with ml = 2nl vertices in load space. Figure (3.2) shows the load domain for two variable loads.
This load domain can be represented in the following linear form:
L = (
P (x, t)|P (x, t) =
nl
X
k=1
µk(t) Pk0 )
(3.28)
3.5 Shakedown analysis of pavements by using the lower bound approach 35
where
µ−k ≤ µk(t) ≤ µ+k, with k = 1, nl and nl = 2.
P10 = p0; P20 = q0.
If the loads traverse along the y-direction, we need to determine the load factor αSD at which shakedown of pavements can occur.
The purely elastic stress σijE(x, t) is defined by (3.4) as the stress would appear in a fictitious purely elastic structure subjected to the same loads as the actual one.
σijE(x, t) =
nl
X
k=1
µk(t) σijEk(x) (3.29) where σEkij (x) denotes the stress field in the reference (fictitious) structure when subjected to the unit load mode Pk0.
3.5.2 Elimination of time variable
It is impossible to apply Melan’s theorem to find directly the shakedown limit in practical computation due to the presence of the time-dependent stress field σijE(x, t) in (3.16). The obstacle can be overcome with the help of the two convex-cycle theorems by K¨onig and Kleiber (1978) [50].
Theorem 6 Shakedown will occur for a given load domain L if and only if it occurs for the convex envelope of L.
Theorem 7 Shakedown will occur for any load path within a given load domain L if it occurs for a cyclic load path containing all vertices of L.
These theorems, which hold for convex load domains and convex yield surfaces, permit us to consider one cyclic load path instead of the entire loading history.
They allow us to examine only the stresses at every vertex of the given load domain instead of computing an integration over the time cycle. Based on these theorems, K¨onig and Kleiber suggested a load scheme as shown in figure (3.6) for two independently varying loads.
Using the above theorems to eliminate time-dependent elastic stress field σEij(x, t), we consider a special load cycle (0, T ) passing through all vertices of the load domain L such as:
P (x, t) =
ml
X
k=1
δ (tk) ˆPk(x) (3.30)
Figure 3.6: Critical cycles of load for shakedown analysis
where ml = 2nl is the total number of vertices of L, nl is the total number of loads (in our case nl = 2), δ (tk) is the Dirac function defined by:
δ (tk) =
( 1 if t = tk
0 if t 6= tk (3.31)
where tk is the time instant when the load cycle passes through the vertice ˆPk. Obviously, the Melan condition required in the whole load domain will be satisfied if and only if it is satisfied at all vertices (or the above special loading cycle) of the domain due to the convex property of load domain and yield function.
This remark permits us to replace the time-dependent stress field by its values calculated only at load vertices. Then the static shakedown theorem becomes:
Theorem 8 If there exist a factor α > 1 and a time-independent residual stress field ρ(x) with
Z
Ω
ρ: D−1 : ρ dΩ < ∞, such that for all loads P(t) ∈ L the yield condition is satisfied,
F [ασE(x, ˆPk) + ρ(x)] ≤ 0 (3.32) then the structure will shake down under the given load domain L.
This leads to the mathematical optimization problem max α
such that: ∂ρij
∂xj
+ bi = 0 in Ω ρijnj = pi on Γ1
F
ασEij(x, ˆPk) + ρij
≤ 0 in Ω ∀k = 1, ml
(3.33)
3.5 Shakedown analysis of pavements by using the lower bound approach 37
3.5.3 Shakedown domain
To investigate the shakedown domain, let us consider the problem of two loads P1 and P2 which has loading domain as in Figure (3.2). Due to proportionality between loads and stresses, the shakedown domain is the biggest rectangle in space of loads, which is defined by:
PiSD = αSDPi0 (3.34)
The residual stress field so determined is common to all corners of the biggest rectangle due to time independence of the residual stress field.
By repeating this procedure for several ratio PP12, we get several loading points which are on the limit of the shakedown domain. According to Morelle (1980) [63], we deduce that the shakedown domain may be determined by the polygonal line joining all obtained points [96], see figure (3.7).
Figure 3.7: Determination of shakedown domain
If the load domain αL shrinks to the point of a single monotone load, limit analysis is obtained as special case.
4 The extended shakedown theorems for non-associated plasticity 39
4 The extended shakedown theorems for non-associated plasticity
In this chapter Melan’s extended theorem proposed by Boulbibane and Weichert (1997) and its proof [9] are rewritten with the usage of the rounded Mohr-Coulomb criterion. Then another Melan’s extended theorem with consideration of both yield and plastic potential conditions is introduced.
4.1 Non-associated behaviour:
Granular soils have non-associated behavior. Due to internal friction and the change of physical state defined by density and volume changes the plastic strain increment deviates from the normality. Therefore, we need an constitutive law of non-associated plasticity flow.
Using associated flow rule to model behaviour of granular materials will lead two following drawbacks. Firstly, the magnitude of the plastic volumetric strains (i.e. the dilation) is much larger than that observed in real soils, and secondly, once the soil yields it will dilate for ever. Real soil, which may dilate initially on meeting the failure surface, will often reach a constant volume condition (i.e.
zero incremental plastic volume metric strains) at large strains.
The first drawback can be partly rectified by adopting a non-associated flow rule, where the plastic potential function is assumed to take a similar form to that of the yield surface (2.31) or (2.30), but with the frictional angle φ replaced by the angle of dilation ψ.