COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China
E2006 Tsinghua University Press & Springer
Effect of Surface Traction on the Shakedown Limits under Moving
Surface Loads
Jim S. Shiau*
Faculty of Engineering and Surveying, The University of Southern Queensland, QLD, 4350, Australia
Email: jim.shiau@usq.edu.au
Abstract: The effect of repeated surface friction on the shakedown limits for an isotropic, homogeneous
Tresca material is examined in this paper using a lower bound finite element and mathematical programming
approach. Based on Bleich-Melan’s static shakedown theorem, numerical solutions are obtained for this
problem and compared with the analytical solution proposed by Johnson (1985). A shakedown map for various
surface friction is prepared for design purposes.
Key words: shakedown limit, lower bound, repeated loadings.
SHAKEDOWN CONCEPT
To explain the shakedown concept in the plasticity design of a continuum, road pavement is one of the good
examples. When a road pavement is overloaded by a heavy vehicle repeatedly moving in a single direction,
it leads to plastic (irrecoverable) deformations on the road surface. A redistribution of stress in the road
pavement, which we cannot see, thus occurs. The stress present in the road pavement following unloading is
known as a “residual stress”. It can be shown theoretically that there is a load magnitude below which a
protective residual stress will develop in the road, and above which the pavement will undergo an incremental
failure. Provided that subsequent loads are less than a certain limit load experienced by the road pavement,
this residual stress in the road pavement offers protection against further accumulation of plastic deformations,
that is, against further rutting. This load is known as the ‘shakedown limit load’ and the protective residual
stresses associated with this shakedown limit load are the optimal residual stresses for the life of the structure.
Psd
Strong Wind
Permanent deformation D r
Residual stresses s
ÏÏÏ ÏÏÏ ÏÏ
Original shape
r
ÏÏÏ
New deformed shape
Figure 1: Simple illustration of shakedown concept (Psd : shakedown limit, D r and s r : permanent
deformation and residual stress associated with the shakedown limit)
Another simple shakedown illustration is shown in Fig. 1. If the tree shakes down after a number of variable
repeated wind loadings, the associated shakedown quantities such as shakedown limit load (Psd ), residual
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stresses (s r), and permanent deformations (D r) are of main concerns in the design of structures subjected to
moving repeated loadings.
MELAN’S STATIC SHAKEDOWN THEOREM AND THE NUMERICAL APPROACH
Melan’s static shakedown theorem states that ‘‘If the combination of a time independent, self-equilibrated
residual stress field s rij and the elastic stresses field ls eij can be found which does not violate the yield condition
anywhere in the region, then the material will shakedown”. In other words, if no such residual stress fields
can be found then the system will not shakedown and plastic deformation will be accumulated at every passage
of the load.
l
Element equilibrium
ę
Shakedown load factor
s eij
ę
D
Displacement boundary
D
D
D
D
D
s rij
Residual stress fields
+
Elastic stress fields
Stress boundary
Element equilibrium
D
D
D
Compatibility
f ǒls eij ) s rijǓ t 0
Discontinuity equilibrium
Figure 2: Graphical representation of Melan’s static shakedown theorem and the finite element application
Fig. 2 shows such a graphical representation of Melan’s static shakedown theorem. Supposing that the
elastic stresses are proportional to a load factor l, the combined stresses are therefore
s tij + ls eij ) s rij
(1)
where l is the shakedown load factor, s eij are the elastic stresses resulting from a unit pressure application and
s rij are the residual stresses.
Melan’s static shakedown theorem provides a simple alternative for estimating the shakedown limit load since
a step-by-step procedure for determining the shakedown limits under complex traffic loads is difficult and
would not be practical in view of high computation costs. The static shakedown theorem enables one to
determine the overall shakedown behaviour of a structure under variable repeated loading. In conjunction with
finite elements and linear programming, this theorem will be employed to predict the shakedown behaviour
of a continuum under plane strain condition. This paper will focus on presenting numerical results for
shakedown limit loads under a moving Hertz contact load with repeated horizontal tractions.
Details of the numerical formulation can be found in Shiau (2001), Shiau and Yu (2000), and Sloan (1988)
and only a brief description will be presented here. Shown in Fig. 2 is the illustration of a finite element
application of Melan’s static shakedown theorem. As indicated, both elastic stress fields and residual stress
fields required by the shakedown theorem are assumed to be linearly distributed across the continua by making
use of the displacement and stress finite elements respectively. By insisting that combined stresses do not
violate the yield condition in the mesh, the calculation of shakedown limits are then considered as a large
mathematical programming problem: the maximisation of the shakedown load factor l subject to the
constraints due to: (1) Element equilibrium; (2) Discontinuity equilibrium; (3) Stress boundary condition; and
(4) Mohr-Coulomb yield constraint.
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PROBLEM DEFINITION AND JOHNSON’S SOLUTION
In order to compare the numerical shakedown limit obtained by the proposed formulation with the analytical
shakedown solution of a free rolling contact derived by Johnson (1962, 1985, and 1992), the contact stress
distribution is assumed to be a form of Hertz contact pressure as shown in Fig. 3. The normal pressure p v(x)
at the pavement surface is distributed semielliptically according to
p v(x) + p v! Ǹ1 * (xńa) 2
(2)
where pv is the maximum contact pressure and a is the semicontact width.
In the case that the surface friction exists, the surface shear traction p h(x) is given by
p h(x) + mp v! Ǹ1 * (xńa) 2
(3)
This implies that a constant coefficient of surface friction m is adopted, where
m+
p h(x)
p v(x)
(4)
x
Pv
a
Ph
p v(x) + pv! Ǹ1 * (xńa)2
p h(x) + mpv! Ǹ1 * (xńa)2
Pv
Ph
m+
p h(x)
pv(x)
X
Layered material
Y
Figure 3: Hertz load distribution for plane strain model
As shown in Fig. 3 , a loading cycle on the pavement surface consists of the loading patch moving from
x + * R to x + ) R. Application of Melan’s quasi-static shakedown theorem requires, in addition to the
elastic stresses, a system of self-equilibrium residual stresses. Because the same loading history is experienced
for all points on the pavement surface, the residual stress distribution must be independent of the travel
direction. It follows that the only residual stresses are s rx and s rz for such a plane strain condition and are
uniform over any horizontal plane. If s rz is chosen as the intermediate stress, we can write the following
equation for using the Tresca yield criterion (Johnson, 1992).
1 NJ(s e ) s r ) * s eNj 2 ) (t e ) 2 x c 2
(5)
x
y
xy
4 x
This equation cannot be satisfied if t exy exceeds c. However, if we choose s rx + sey * sex , it can just be satisfied
with t exy equal to c. Thus, the limiting conditions for shakedown to occur in the solid is possible when the value
of t exy is a maximum. The maximum elastic shear stress under the Hertz contact stress distribution is given in
Johnson (1962) where (t exy)max + 0.25p v occurs at x +" 0.87a and y + 0.5a. This result has also been verified
by using the displacement finite element method in this thesis which shows that (t exy)max + 0.2497p v in Fig. 4. It
therefore gives a lower bound to the shakedown limits such that
p vńc y 4.00
(6)
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0.87a
(t exy) max+0.2497
0.5a
-.01114
.017845
.046829
.075813
.104797
.133781
.162766
.19175
.220734
.249718
m + 0
Figure 4: Contour of elastic shear stress txye ( Hertz contact distribution)
Johnson(1992) also described a simple upper bound solution by applying Koiter’s kinematical theorem. A
mechanism of incremental collapse comprising simple plastic shear along a plane y=y0 parallel to the outer
surface of the solid is proposed. If the increment of plastic tangential displacement is Du px, then the work done
by the elastic stresses is t exy!Dupx and the internal work dissipation is c!Du px. An optimum upper bound on the
shakedown limit is thus found by taking y0 at a depth of (t exy)max + 0.25p v which gives
p vńc v 4.00
(7)
Since the lower and the upper bounds are identical , as shown in equations (6) and (7), they represent the true
shakedown limit for such a plane strain model under moving repeated loading.
x
Hertz contact pressure
lP v
s rn + t r + 0
tr + 0
Number of nodes = 936
Number of elements = 312
Number of discontinuities = 444
Number of nodes = 673
Number of elements = 312
Stress−based finite element mesh
Displacement finite element mesh
Figure 5: Finite element symmetric mesh for shakedown analysis (fan type)
A fan type of finite element mesh used for both elastic stress field and residual stress field is shown in Fig. 5.
In the case that only normal stress is applied (p hńp v + 0), a symmetric mesh can be adopted. The displacement
finite element mesh consists of 312 quadratic elements and 673 nodes while the stress finite element mesh
consists of 312 linear stress elements and a total of 936 nodes. The total number of discontinuities for the
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stress-based mesh is 444. The advantage of using a fan mesh arrangement is that more elements can be used
to pass the singular point, where the jump in loading condition is obvious. This also has advantage over the
derived elastic stresses around this singular point which is expected to yield a less accurate stress output. The
shakedown limit obtained from this particular study for a purely cohesive material with p hńp v + 0 is 3.953
which is 1% less than that reported in Johnson 1962 and 1985.
EFFECT OF SURFACE FRICTION
The effect of surface friction on the shakedown limits for an isotropic, homogeneous cohesive soil is presented
in Fig. 6. It can be seen that the dimensionless shakedown limit decreases dramatically with the increase
in the coefficient of surface friction m. This is mostly due to the existence of high elastic shear stresses in the
near surface which tends to cause surface shear failure when the value of m is high. Using the same shakedown
formulation, the elastic limit loads, as shown in Fig. 6, are obtained by insisting that no residual stresses
exist in the media. The differences between these two curves thus indicates the benefit of shakedown
phenomenon under repeated loadings.
4.5
Subsurface Failure
4.0
Surface Failure
3.5
lp v
c 3.0
Shakedown Limit
2.5
2.0
Elastic Limit
1.5
1.0
0.5
0.0
0.0
0.37
0.2
0.4
0.6
0.8
1.0
Coefficient of Surface Friction
m
Figure 6: A Shakedown Map indicating the effect of the coefficient of surface friction m upon
dimensionless shakedown limits
It was previously shown that, for shakedown to occur, the orthogonal shear stress (t exy)max must not exceed the
shear strength c at any point in the stress field. The position and magnitude of (t exy)max will have to be evaluated
for different values of surface friction. These have been done by using displacement finite element analysis
and are presented in Figure 7 and Figure 8 where stress contours of (t exy)max are plotted for different values of
m.
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(t exy)max + 0.253
-.252849
-.196661
-.140472
-.084283
-.028094
.028094
.084283
.140472
.196661
.252849
(a).m + 0, lp vńc + 3.953
(t exy)max + 0.317
-.192467
-.135853
-.079239
-.022626
.033988
.090602
.147216
.203829
.260443
.317057
(b).m + 0.2, lp vńc + 3.143
(t exy)max + 0.365
-.157461
-.099386
-.041311
.016763
.074838
.132913
.190988
.249063
.307138
.365212
(c).m + 0.37, lp vńc + 2.731
Figure 7: Elastic shear stress contour txye for various surface frictions (I)
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(t exy)max + 0.381
-.145793
-.087231
-.028669
.029893
.088455
.147017
.205579
.264141
.322702
.381264
(d).m + 0.4, lp vńc + 2.617
(t exy)max + 0.567
-.100066
-.025894
.048277
.122449
.196621
.270793
.344965
.419137
.493309
.567481
(e).m + 0.6, lp vńc + 1.762
(t exy)max + 0.940
-.050742
.059331
.169404
.279477
.389551
.499624
.609697
.71977
.829843
.939916
(f).m + 1.0, lp vńc + 1.064
Figure 8: Elastic shear stress contour txye for various surface frictions (II)
Further examination on the elastic shear stresses from these figures shows that the location of maximum shear
stress moves from a depth of y=0.5a when m + 0 to the pavement surface when m is approximately equal to
0.37. This indicates a type of transfer from subsurface failure to the surface failure. When m + 0, the direct
elastic stress components are symmetric and the elastic shear components are antisymmetric about the central
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axis. These figures also show that the effect of a surface tangential stress ( m u 0) is to increase one of these
peak values (t exy)max and, at the same time, to decrease the other.
Another interesting study on the boundary effect has also shown in Fig. 9. It indicates that the depth (H)
of finite element mesh has no effect on the shakedown capacity when the coefficient of surface friction
( m + p hńp v) is high. This is because the shakedown limit is governed by the surface stresses when m is high,
as explained earlier.
6
5
L/B=3.0
ph/pv=0
ph/pv=0.4
4
lp v
c
3
PV
2
PH
1
B
H
0
H/B
0
1
2
3
4
5
6
7
c, f + 0
L
8
Figure 9: Converged results on H/B for dimensionless shakedown limits
Smith and Liu (1953) computed the position and magnitude of (t exy)max with increasing surface friction and these
results are presented in Table 1. It can be seen from this table that as m increases, (t exy)max also increases, but
its location slowly decreases in depth (y/a). However, the decrease in x position is not significant from their
study. It is worth noting that when m reaches 0.367, there are two locations for the maximum values of (t exy)max.
They are: x/a=0, y/a=0 and x/a=0.858, y/a=-0.366. Similar results are also obtained from displacement finite
element analysis which have been shown in Fig. 7 and Fig. 8.
Table 1 Computed maximum orthogonal shear stress (Smith and Liu, 1953)
m
x/a
y/a
(t xy) max
0
+0.866 , -0.866
-0.5
0.25
0.1
0.866
0.281
0.2
0.864
-0.452
-0.415
0.3
0.861
-0.384
0.345
0.367
0.858
-0.366
0.367
0.367
0
0
0.367
0.312
It is well known that the first yield (elastic limit) is reached at a point beneath the surface. The existence of
the surface shear forces will introduce a new state of stress at the surface. When the coefficient of surface
friction exceeds a certain value, yield may begin at the surface rather than beneath it (Johnson, 1985). It has
been demonstrated that this critical value of m is approximately equal to 0.37 from our displacement finite
element study. It is also noted that for cases m u 0.37, the critical condition for shakedown has moved to the
surface and is then controlled by the surface stresses. Thus, it may be concluded that the shakedown limit load
is not significantly different from the elastic limit load for high values of m. Fig. 6 showed such a state where
elastic limits are rather close to the shakedown limits at high coefficient of surface friction m. This may imply
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that the protective residual stresses may not be developed in the case where high surface shear stresses exist.
Further study of these protective residual stresses with varying m would be interesting.
CONCLUSION
The effect of surface friction on the shakedown performance of a continuum under repeated moving surface
loads have been examined in this paper. Results have shown that the currently numerical approach predicts
accurate shakedown limits under a Hertz moving surface load, as compared to Johnson’s analytical approach.
Future investigation on the residual stress distribution is recommended.
Acknowledgements
The author would like to acknowledge Professor Scott Sloan, Professor Hai-Sui Yu, and Dr Andrei Lyamin
for their guidance during the period 1998-2003 at the University of Newcastle, Australia.
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