hase halfspace analysis

HASE half-space analysis 2011 page 1

SOFiSTiK

HASE

Half-Space Analysis

SOFiSTiK AG

www.sofistik.com


1. Half-space analysis

1.1 Why to use half-space analysis

One of the ways to model the static soil-structure interaction is to use 3D finite elements – figure 1 on the left.

This however requires a huge number of soil elements which in many cases can sever times surpass the

number of elements needed for modeling of the structure itself, which is usually the main part of interest for

structural engineers. It is therefore very useful to adopt a sub-structuring technique in which the structure will be

represented using standard finite elements, while the soil will be modeled semi-analytically using the half-space

theory – figure 1 right. The connection between the two substructures will be assured by stiffness/flexibility

coefficients at the soil-structure interface. This would then allow a more detailed discretization of the foundation

slab and the structure itself.

Figure 1: necessary mesh for half-space soil structure interaction compared with full 3D bric model

1.2 Stiffness coefficient method

This model takes into consideration shear deformations within the soil. A point load on the surface produces

stresses and deformations in the surrounding soil and causes the surface to form a smooth depression centered at

the load (figure 2c). Constant surface compression causes a rounded depression (cavity) which leads to stresses in

the building foundation and in the structure itself (figure 2a).

Figure 2: (a) Uniform load on L-shaped bottom slab producing settlement cavity on a half-space surface (green

grass added only for visualization) (b) Bore profile (c) Pressure trajectories in the half-space subjected to point load

(b) (a) (c)

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Full 3D BRIC model

3D half-space model

half-space


1.3 Creating half-space stiffness matrix

The starting point of determining the half-space stiffness matrix is the assembly of the flexibility matrix of the half-

space (soil). Flexibility coefficient δik describes the displacement vi of a point i due to a unit force Pk acting on the

point k in the half-space.

Figure 3: (a) Stresses in vertical direction σz due to a point load P acting on the surface of the half-space (b)

Stresses in the vertical direction σz due to a point load acting inside a half-space at the depth z

In a first step the stress in the soil due to the point load Pk acting on the surface of the half-space is calculated with

a simple half-space formula according to Smoltczyk [1], describing pressure trajectories (figures 2c and 3a):

3

, 3

3

2

kz i

P z

Rσ

π

⋅ ⋅= , (1)

where x, y, and z represent the coordinates of the observed point within the half-space, measured from the point of

load introduction at the half-space surface, while 2 2 2R x y z= + + .

This distribution of stress σz divided with the soil stiffness modulus Es is integrated over the depth of the soil to

give the expected deflection vi due to the force Pk . This technique allows taking into account layered soil as shown

in figure 2b as well as varying soil conditions in ground view (extended half-space). Because equation (1) is

determined under the assumption of the uniform half-space, the shear behavior of a layered soil cannot be taken

into consideration completely correct. Nevertheless the results are acceptable also for a layered soil with varying

stiffness in depth and ground view.

The same method can be used to calculate displacements (or the flexibility coefficient δik) due to the loads

acting within the half-space. The only difference is in the formulas for the calculation of stresses, which can also be

found in Smoltczyk [1]:

( ) ( ) ( ) ( )( )3 2

, 2 5

3 13 4 5

8 (1 )

k

z i

P

Rσ ς ς ν ς ς ς ς ς ς ς ς

π ν ρ

= − + − + − + − + −

( )( )

( )3

7 3

1 210 11

3

νςς ς ς ς ς

ρ ρ

− + + − − −

, (2)

where /z Rς = , /z Rς = and /R Rρ = . The meaning of z , R and R is explained in the figure 3b.

(x,y,z) (x,y,z)

x

y

x

z

y z

P

P

R z

r

R

Ř

σz σz

z= Ŝ

Ŝ

(a) (b)


When the flexibility matrix is assembled, by inversion the stiffness matrix of the half-space can be obtained,

connecting all nodes of the soil-structure interface (nodes of the bottom slab) in a huge stiffness bubble.

1.4 Linear finite element analysis

Once computed, the aforementioned soil stiffness matrix can then be used in a standard finite element analysis.

The following summarizes the algorithm for total linear analysis run:

1. Calculation of the soil−profile distribution in the plan view

2. Calculation of the flexibility matrix

3. Inversion and storage of the stiffness matrix

4. Static calculation including both soil and structure

5. Post-processing for stresses in the soil (also in deeper visualization cutting planes inside the soil)

Figure 4: (a) Deformation and stresses in the building on the half-space (b) Possible visualization cut in the half-

space

1.5 Piles in the half-space

Using extended formula given in Smoltczyk [1] (eqn. (2)) the effects of loads acting within the half-space can be

taken into account. This makes it possible to incorporate piles in the treatment of the half-space. As shown in figure

5, the pressure trajectories of the numerous internal and external forces Pk overlap in the half-space leading to a

complex interaction of piles with each other and with the foundation slab. This is called combined pile-slab

interaction. Figure 4b also shows the effect of a pile foot as a “hot spot” inside the half-space.

Figure 5: Pile-slab interaction – superposition of pressure trajectories in the half-space

(b) (a)


2. Nonlinear half-space contact analysis

Solving the global system consisting of bottom slab, building and half-space with the real external loads, we first

obtain linear displacements, forces and stresses. These stresses can be checked and reduced if necessary in a

nonlinear iteration.

2.1 Foundation slab soil pressure First nonlinear effect that is taken into consideration is the reduction of boundary soil pressure under the foundation slab, exceeding defined maximum limit. Linear analysis of a uniformly loaded bottom slab causes concentration of the stresses at the boundary as shown in figure 6.

Figure 6: Uniformly loaded bottom slab: red=linear result, blue=nonlinear result with yield plateau

2.2 Method of nonlinear residual force iteration

The method for the half-space contact iteration differs from a common finite element nonlinear iteration where a

nonlinear answer of the element force to a given displacement can be used directly.

In a half-space analysis presented here, it is necessary at the end of the each iteration to guarantee that the half-

space displacements correspond to the nonlinear reduced contact forces. A reduction or limitation of a force in one

contact node causes a change in all other half-space nodes. So in order to preserve this global relationship

between interface nodes the reduced contact force vector is multiplied with the half-space flexibility matrix, resulting

in a displacement vector. Comparing this displacement vector with the actual finite element displacements we can

extract correction forces from the difference between these two vectors. At the end of a convergent iteration these

displacements can be plotted as in case of example in figure 7 showing the plastic sinking of the boundary of the

foundation slab.

Figure 7: blue=bottom slab, red=theoretic half-space surface, difference = plastified (squeezed out)

2.3 Pile foot limitation and pile skin friction

Similarly to the soil pressure underneath the foundation slab, it can be checked if the contact forces of piles

exceed allowable values at the pile foot and along the pile (pile skin friction). A plastic yielding of the pile

reactions will cause stress in the foundation slab bedding to increase. In the example below (figure 8), the

yielding of skin friction causes pile foot force to increase.

Halfspace surface

Stiff bottom slab Squeezed out

Yielding plateau

Linear result


Figure 8: Linear (a) and nonlinear (b) pile reactions (20 kN/m maximum friction allowed in the upper part)

2.4 Horizontal pile forces

Assuming a horizontal half-space stiffness to be affine to the vertical stiffness, horizontal forces can also interact

over the half-space. This approach is not an optimum but it is much better than individual global fixed piles

without any pile interaction. The horizontal pile bedding can as well be checked against an allowable horizontal

bedding force. With this approach it is also possible to take into consideration inclined piles in a nonlinear

analysis (figures 9 and 10).

Figure 9: Examples for piles in half-space with horizontal effect

Figure 10: Higher nonlinear (b) pile bending due to horizontal soil yielding on upper pile part

Soil resistance yielding

P

axial

lateral

(b) (a)

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mantle yield level 20 kN/m

pile foot


3. Example pile-slab interaction

The following example is taken from Katzenbach et al. [2], and it shows the effect of pile-pile as well as the pile-

slab interaction. It takes into consideration all the aforementioned non-linear effects – maximum pile foot force,

maximum skin friction and maximum soil pressure underneath the slab.

Two piled raft systems are analyzed – one with 25 and the other with 9 piles connected to the pile cap. Pile

elements are modeled with 1D beam elements, while the slab is modeled with the 2D quadrilaterals. Piles have

circular cross section with diameter D = 1.5 m and length l = 20xD. Slab is quadratic in base with side length a =

18xD and thickness d=1.0 m. Piles are distributed over regular grid with the distance of e=3xD for the system with

25 piles and e=6xD for the system with 9 piles.

Structure is made of reinforced concrete of class 30 according to DIN 1045. Soil is modeled as half-space with

stiffness modulus of 0.6 GPa and Poisson’s ratio of 0.25. Maximum soil pressure underneath the slab is set to be 5

MN/m2. Maximum pile skin friction is constant along the pile and is equal to 60 kN/m2, while the maximum pile foot

force is chosen to be 3000 kN.

The structure is subjected to uniform constant load acting on the top of the slab with the total resultant value of 200

MN. Figure 11 shows the differences in the deformation of the pile-slab system with 25 piles and the one with

only 9 piles.

Figure 11: Deformation of the pile-slab system subjected to uniform load of total intensity RTOT = 200 MN (scale

= 60); (a) 25 piles, e/D = 3, maximum displacement 6.3 cm (b) 9 piles, e/D = 6, maximum displacement 8.6 cm

Figure 12 shows the distribution of the normal force along a pile length for corner, edge and central pile of the

pile-slab system with 25 and 9 piles. As can be seen from these figures, in the case of the 9-pile system the

distribution of the normal force is much more balanced. The differences are less than 3%. In the case of the

pile-slab system with 25 piles, differences are significant. As expected, the largest difference is between the

central and corner pile with the factor of more than 2. Both systems exhibit nonlinear behavior. Results show

that the maximal skin friction in both systems is exceeded in the lower parts of the piles. However, while for the

pile-slab system with 9 piles the maximum foot force of 3000 kN is reached in all piles, in the case of a 25-pile

system none of the piles exhibit this behavior. The soil stress underneath the slab in both cases is way below

the chosen limit of 5 MN/m2.

(a (b


Figure 12: Normal force distribution for the corner, edge and central pile in the pile-slab system with (a) 25 piles

and (b) 9 piles

Figure 13 depicts the force-displacement behavior of the combined pile-slab system. Load is varied between 0

and 3xRTOT, where RTOT = 200 MN. Vertical displacement of the center of the slab is recorded and plotted

versus total load. As mentioned before, both of the systems exhibit nonlinear behavior, however there are

some differences. In both systems nonlinearity starts relatively early - first the maximum skin friction is

exceeded in the lower parts of the piles and it progresses successively towards the surface. In the case of 9-

pile system this “yielding” occurs earlier, under smaller total force than for the case of 25-pile system. Next the

maximum pile foot force starts to “yield” – for the 25-pile system this “yielding” starts at 1.22 RTOT, while for the

9-pile system it starts at 0.70 RTOT. From the figure it appears that in this case the nonlinear behavior of the

piled raft system is mostly dominated by the “yielding” of the pile foot force.

Figure 14 shows how the part of the total resultant load transferred by the piles to the ground depends on the

settlement of the center of the foundation and on the total applied load.

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(b)


Figure 13: Force-displacement curve of the pile-slab system subjected to uniform load. R represents the total

resultant load applied at the top of the slab, while the displacement is the displacement of the center of the slab.

Figure 14: Dependency of the contribution of the piles to the total resistance of the piled raft foundation on (a)

displacement of the center of the slab and (b) total load applied at the top of the slab

Conclusions

When analyzing a soil-structure interaction it can be very useful to concentrate on the meshing of the

structure and not to avoid discretizing the soil with large number of volume brick elements, which can

sometimes several times surpass the number of elements needed for the modelling of the structure itself.

This work shows that it is possible to model the static behaviour of soil by representing it with

flexibility/stiffness matrix at the soil-structure interface based solely on the half-space theory. This paper also

shows that it is possible to incorporate nonlinear effects using this half-space technique. The nonlinear half-

space contact allows not only vertical but also horizontal stress interaction in the soil and redistribution of

loads from piles to the slab and vice versa.

References

[1] Smoltczyk U., Grundbau Taschenbuch, 3. Edition, Part 1, Verlag Wilhelm Ernst & Sohn, Germany,

1980

[2] Katzenbach R., König G., et al., Richtlinie für den Enwurf, die Bemessung und den Bau von

Kombinierten Pfahl-Plattengründungen (KPP), Fraunhofer IRB Verlag, Germany, 2000

hase halfspace analysis

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