4.16 Plane problems in soil mechanics

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It is easier to work in soil mechanics with plane models than with full 3-D

models. In this context there are three topics to be discussed.

Self-equilibrated stress states and primary load cases

In soil mechanics particular attention must be paid to the various construction

stages, because in nonlinear analysis it must be possible to define a primary

stress state to assess the stress history correctly. Elements are removed or

their stiffness is reduced, if for example, an injection is washed out or a certain

segment of the soil is defrosted. Then the vanishing stresses generate loads. To

handle these stresses in a program it must be possible to identify the stresses

with certain loads, which are so tuned that the sum of these loads cancels

in the undisturbed zones, while they produce true loads at the edge of the

disturbed zone.

4.16 Plane problems in soil mechanics 387

Fig. 4.53. Primary stress state and additional load

The nodal forces of the primary state are determined using the principle

of virtual displacements. Let us assume that as in Fig. 4.53, the stresses due

to the gravity loads are zero at the upper edge, 20 kN/m2 at the interface

between the two elements, and 40 kN/m2 at the lower edge. At the centers of

the elements the stresses are 10 kN/m2 and 30 kN/m2, respectively. Hence in

the primary state there results a pair of opposing forces: at the upper edge a

force of 10 kN and at the lower edge one of 30 kN; the length of the elements

is assumed to be 1.0 m. If the gravitational forces of 10 kN are added to all

element nodes, the resultant forces vanish at the upper nodes, while at the

bottom the nodal forces equal the total load.

If the gravity load had not been applied, the loads obtained would have

pointed upward and been of the same magnitude as the external load, hence

the displacements and stresses in the primary stress state would be zero.

There is a peculiar effect with regard to the horizontal stresses. Because

these stresses do not enter the equilibrium conditions directly they can only

be recovered if the vertical stresses are multiplied by ν/(1−ν). But very often

the so-called lateral pressure ratio of the soil will not agree with Poisson’s

ratio because of geologic preloads and possible plastifications. Even after a

complete removal of the vertical load the soil will not be stress free.

Settlements

A strange effect also exists with regard to the calculation of the settlements

under a footing: the more the mesh extends in all directions the greater the

settlements.

This effect has to do with the behavior of the natural logarithm, ln r. When

a single surface load P = 1 is applied at the edge of the elastic half-plane,

the deflection on the left- and right-hand side of the point load essentially

resembles ln r, that is, at the base of the source point r = 0 the deflection is

infinite, w = −∞. (The y-axis points upward, the load P points downward.)

388 4 Plane problems

Fig. 4.54.

edge of the elastic

half-plane

The strange thing is that there is also a singularity at the other end: moving

away from the source point, the absolute value of the deflection |w| will first

decrease, and then after a while it will increase again, and will tend to +∞

as r→∞.

Let us assume that a load p(x) = p(x1, x2) acts at the edge (x2 = 0)

of the half-plane in the interval −1 ≤ x1 ≤ +1 (see Fig. 4.54). The vertical

displacement of the soil at a point x = (x1, x2) is, if we simplify somewhat

and concentrate on the essentials,

v(x) =

_ +1

−1

1

2 π

ln r p(y) dsy . (4.122)

From the standpoint of a mole that burrows into the soil, the edge load at

the surface more and more resembles a point force, so that at a large enough

distance r from the surface the deflection becomes

v(x) =

1

2 π

ln r

_ +1

−1

p(y1, 0) dy1 =

1

2 π

ln r P ,

where P is the resultant force of the edge load.

This means that in any system of loads that are not self-equilibrated, any

increase in the radius of the mesh will increase the displacements, and if the

radius tends to infinity, so will the displacements.

For this tendency to prevail to the very end the lateral parts of the mesh

must not be chopped off, because otherwise the sheet piling (u = 0) might

act as an abutment, and the load would be carried by an arch which develops

beneath the footing. Therefore an unimpeded expansion of the soil must be

possible.

Various remedies overcome this problem: firstly one could simply set a

limit on the mesh size and set the vertical displacement to zero at the bottom

of the mesh, v = 0. Secondly one could modify the load to make it selfequilibrated.

Thirdly one could let the modulus of elasticity increase in the

vertical direction. This would lead to finite displacements.

Distributed

load at the

4.16 Plane problems in soil mechanics 389

Fig. 4.55. Building bricks

Discontinuities

A foundation slab and the soil have different material properties. This alone

may cause trouble. Putting a building brick on top of a second brick (see

Fig. 4.55) looks harmless, but at the interface between the two bricks, stress

peaks and even singularities may be observed. The magnitude of the stress

peaks depends on the stiffness ratio of the two bricks. The extreme values are

marked by an infinitely flexible slab and a perfectly rigid slab.

In the first case—if the modulus of elasticity of the lower brick is distinctively

greater—the pressure on the lower brick will be nearly constant and

the singularity will be hardly noticeable. Outside the loaded region, the vertical

stress abruptly drops to zero, but this discontinuity soon gives way to a

continuous stress field near the interface.

But the stress discontinuity at the interface between the two blocks cannot

be modeled, because the condition that the two blocks have one edge in

common enforces a constant strain in both elements over the whole length,

which leads to a conflict; see Fig. 4.55.

In the second case (rigid punch on a half-space) the soil pressure becomes

infinite at the edges of the punch. To see these stress peaks, the mesh must

be sufficiently refined. The problem was analyzed with a coarse and a fine

irregular mesh. Because of the minor irregularities in the layout of the mesh

(Wilson’s element), the stresses on the left- and right-hand side were slightly

different, even though the system and the load were symmetric. Asymmetries

in the displacements were not noticeable. Different ratios of the modulus of

elasticity were tested. Table 4.13 lists the vertical stresses at the outer contact

nodes under a total load of 100 MPa.

The results are remarkable. Even though no inferior elements were used,

the bandwidth of the results is surprising. In the case of a soft slab, the stress

at the extreme node of the interface is 50 MPa, which is simply the average

value of 0 and 100. The more the mesh is refined, the greater the stresses

become. Even on the finest mesh, there are noteworthy deviations between

the results for the two corner nodes on the left- and right-hand side. But it

would make no sense to put more effort into the analysis in order to drive

the stresses towards infinity. To assess the crack sensitivity, the results of a

medium-sized mesh are in general sufficient.

390 4 Plane problems

Table 4.13. Vertical stresses left/right (MPa) at the extreme nodes in the contact

area for different ratios of η = Eupper/Elower

elements η = 0.01 η = 0.1 η = 1.0 η = 10 η = 100

521 44 / 55 55 / 67 102 / 107 138 / 133 143 / 136

884 39 / 72 56 / 94 152 / 179 226 / 216 222 / 208

2308 52 / 61 88 / 103 303 / 341 404 / 459 321 / 398

Fig. 4.56. The prinaround

the tunnel

Tunnels

Shallow tunnels are designed to carry the full weight of the soil. But with increasing

depth—beginning at a distance of about a full diameter, and depending

on the quality of the soil material or the rock—the stresses are redirected

around the tunnel (see Fig. 4.56), so that the load is carried by the rock, and

the tunnel shell simply provides shelter for the traffic passing through the

tunnel.

Modeling the soil with a 3-D mesh and simulating the complete construction

process by successively deactivating the tunnel elements and activating

new shell elements (tunnel lining) brought into the tunnel results (even in the

case of a short tunnel) in a very large system of equations. After each step,

this system must be rearranged and solved iteratively, due to the nonlinear

effects. Therefore a conventional 3-D analysis will be restricted to short sections

of the tunnel at the very front of the excavation, near crossings, and

crevices.

Thus, the stress analysis of tunnels nowadays is mostly based on plane

models, where by suitable modifications one also tries to take into account

the redistribution of the internal forces in the direction of the tunnel axis.

The various construction stages and the sequence of excavations complicate

cipal stresses flow

4.16 Plane problems in soil mechanics 391

Fig. 4.57. 2-D analysis: a) global mesh, b) two excavation steps, c) cross section

the numerical analysis even of the plane model, because (see Fig. 4.57) the

front of the tunnel is divided into a variety of different cross sections, and the

nonlinearity of the strains and stresses in the rock near the tunnel front must

be considered.

In an FE model, this sequence of events is normally modeled by an iterative

procedure, where a new step in the analysis starts with the stress state of the

392 4 Plane problems

Fig. 4.58. 3D-model of the excavation

previous steps, and where the stiffness of the elements is reduced according

to the construction process by a suitable empirical factor 0 < α < 1.

The alternative to a plane model is a 3-D model with a so-called excavation

by driving, where the excavation sequence is modeled by a special 3-D mesh

(see Fig. 4.58). Imagine a sequence of, say, 20 tunnel “slices” (cross sections

with a thickness of about 1 m) that wander as a package through the rock.

Because the mesh is moving with the excavation—from the point of view of

the mesh, the rock is moving—the mesh can be kept relatively small. The

iteration proceeds as follows:

1. First the primary stress state in the untouched rock is calculated. Next the

sequence of excavations follows and the complete tunnel lining is activated.

The first step in this sequence yields unrealistic results.

2. By iteratively rearranging the results the stresses in the rock are moved

towards the tunnel opening.

3. This modified stress state then serves as the starting point for the next

calculation. This reanalysis is repeated until the modifications become

negligible.

In this manner, the actual front of analysis advances through the rock in

concert with the excavation. The unrealistic results of the first step move

towards the tunnel opening and leave the rock zone.

The advantage in comparison with a conventional 2D-analysis is that both

structural behavior in the longitudinal direction and the relaxation of the rock

are accounted for by the analysis.

4.18 Mixed methods 393

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