structural elements and interfaces

Structural Elements and Interfaces

PLAXIS FINITE ELEMENT CODE FOR SOIL AND ROCK ANALYSES

Structural elements Numerical analysis in geotechnical engineering generally require themodelling of structural elements to simulate the behaviour of a structurebeing involved in the design. Structural elements could be modelled as continuum elements and thiswould make the analysis very accurate but not convenient from the practicalpoint of view because the difficulty in generating the FE model.Such kind of approach would be impossible when the number of structural


Such kind of approach would be impossible when the number of structuralelements becomes relevant because of the extremely high number ofelements needed and degrees of freedom introduced, thus making theanalysis extremely heavy from the computational point of view.Furthermore, the output of structural forces, such as shear and normalforce and bending moment, can be obtained from volume elements but onlyafter integration of stresses which make the design process laborious.

Structural elements in PlaxisIn order to overtake those difficulties, special finite elements have beendeveloped, in which the input is simplified and structural forces can beobtained straightforwardly as output.

Another kind of special element which is very important in geotechnicalanalysis is the interface.As it is often assumed in practice, the mechanical behaviour at theinterface between soil and a structure, it is not realistic if modelled under the


interface between soil and a structure, it is not realistic if modelled under thehypothesis of perfect adhesion. Hence, it is often assumed that the contact behaviour is linear elastic untila failure stress is reached, and the behaviour is perfectly plastic henceforth.Although in this Lecture special focus is given to those elements which areimplemented in PLAXIS finite element code, the formulation of structuralelements is general and similar elements can be found in other commercialFE codes

Structural elements in Plaxis

Plates and shells (walls, floors, beams, tunnels)

Anchors


Geogrids (geotextiles)

Interfaces

Plates and shells

3 or 5 noded line elements

3-noded beam elements are used in combination with 6-noded triangular elements for the continuum, whereas 5-noded beams are used with 15-noded elements

3 degrees of freedom per node (horizontal and vertical displacement and rotation)

Elastic or elasto-plastic behavior


Elastic or elasto-plastic behavior

The out-of-plane dimensions depend on the type of analysis: it is a unit thickness in plane strain, 1 rad in axisymmetry

To model walls, floors, tunnels

Input parameters for plates

Flexural rigidity (b=1 m)

Normal stiffness (b=1 m)

Element thickness where d is the real (physical)structure of the sheet.

12

3 bhEEI =

bhEEA =

12 EId hEA

=

The flexural and axial rigidity of the structural element are input parameters:

The input of stiffness parameters is


b

h hb

b = 1 m in plane strainb = 1 meter in axisymmetry

The input of stiffness parameters is completed by Poissons ratio .

Plate weights Compensate for overlap:

( )concrete soil realw d = For soil weight use:

unsat above phreatic level sat below phreatic level


Plate weights for tunnels

rinsideroutside

r

dreal

Special curved elements are specifically designed for tunnels.


Overlap is only for half the lining thickness

lining soil

( )outsideinside rrr += 21( ) ( )realsoilrealconcrete ddw 21=

Boundary conditions PLAXIS offers several alternatives depending on the particular kind of fixity that has to be modelled. By default, a structural element is free to rotate even if intersecting the external boundary of a mesh, unless a boundary condition is explicitly specified. In order to prevent horizontal displacements on the left and right side of the mesh and horizontal and vertical displacements at the bottom, PLAXIS offers the standard fixities option. Furthermore, if it is required to prescribe some extra condition, e.g. prevent


Fixed rotation

X

Y

0

1 2

3

4 5

6

Free rotation

plate Rotation fixed at (partly) fixed boundaries Rotation free at free boundaries

Furthermore, if it is required to prescribe some extra condition, e.g. prevent rotations at the one end of the beam, the Moment fixity option has to be selected and applied to the desired end.

Hinges

5

6

7

8

Rotationspring

Spring data:StiffnessMin/Max moment


Hinged connection

Rigid connection

spring

Determination of eff. plate weightsMaterial parameters:

E = 20106 kPa = 0.2 = 24 kN/m3

Plate:d = 0.4 m

12

3 bhEEI =

bhEEA =

EAEIhd 12==


I = bd3 / 12 = 1(0.4)3 / 12 = 5.3310-3 m4A = bd = 10.4 = 0.4 m2

wnet = wgross - soil d

EI 1105 kNm2/mEA 8106 kN/mwnet= 0.4 24 - 18 0.4 = 6.0 kN/m2

Wall:d = 0.2 m

I = 1(0.2)3 / 12 = 6.6710-4 m4A = 10.2 = 0.2 m2

Determination of eff. plate weights


A = 10.2 = 0.2 m

wnet = wgross - soil d

EI 0.13105 kNm2/m

EA 4.0106 kN/m

wnet = 0.2 24 - 18 0.2 = 3.0 kN/m2

Determination of soil stiffness

Stresses at reference point

Initial stress:' = 5 18 = 90 kPa (initial stress)


Initial pre-consolidation stress:c = 5 18 + 20 = 110 kPa ( pre-consolidation)

c = max. stress that ref. point has ever experienced in the past5 m = depth of reference point (before excavation)18 kN/m3 =unit weight of the soil 20 kPa = this is an assumed pre-overburden pressure (POP)

at soil surface, characteristic for the region considered

Determination of soil stiffness

After excavation:(0) = 2.5 18 = 45 kPa

(0) = real vertical stress after excavation in reference point2.5 m = depth of reference point after excavation18 kN/m3 =unit weight of the soil

Stresses at reference point


18 kN/m3 =unit weight of the soil

After loading:(2) = 45 + 6 + 125 = 176 kPa

(2) = real vertical stress after loading45 kPa = ((0) see above)6 kPa = weight of the floor125 kPa = 2 200 kN + 2 300 kN (point loads) / 8 m (width of floor)

Parameters for Mohr-Coulomb model

0

0.5

1

1.5

0 50 100 150 200 250 300 350 400 (kPa)

Assume the sample originates from the reference point. Unloading from ' to 0 and reloading from 0 to ' does not give deformation (elastic


2

2.5

3

3.5

(

%

)

give deformation (elastic behaviour)From unloading/reloading curve, from ' to c (from 90 kPa to 110 kPa):

From primary loading curve from c to 2 (from 110 kPa to 176 kPa):

Parameters for Mohr-Coulomb model

0

0.5

1

1.5

0 50 100 150 200 250 300 350 400 (kPa)

(

%

)Combined:


2

2.5

3

3.5

%1.121 =+=

8690176 ==

7800%1.1

86==

=

oedE

4.0'=

kPa

( ) 364078002.06.04.1

'21'1'1

' =

+= oedEE

kPa

Fixed-end anchors Elastic-perfectly plastic spring elements are provided: one end is fixed (no displacements allowed), the other end is connected to one node of the mesh. These elements can be useful when modelling a symmetric problem, like an excavation supported by props, in which, for symmetry reasons, one end of the anchor is prevented to move. Input parameters are the axial rigidity EA, the spacing Lspacing and the maximum axial force which can be applied to the anchor.


Fixed-end anchors For the cases in which there is no logical reason to assume that the end will stay fixed, PLAXIS provides node-to-node anchors, in which two nodes of the mesh are connected by means of elastic-perfectly plastic spring elements. Plate anchor is a typical examples of application of node-to-node anchors. An anchor designed to support a diaphragm wall is connected to a vertical plate which mobilizes passive thrust to increase the stabilizing forces. A cofferdam consists in a dam obtained by enclosing a portion of the ground between two walls. This is yet another typical application of node-to-node anchors. Input parameters are the axial rigidity EA, the spacing (Lspacing) and the maximum


Input parameters are the axial rigidity EA, the spacing (Lspacing) and the maximum axial force which can be applied to the anchor. A pre-stress can be assigned to the anchor by a double-click on the structural element.

Geogrids Geogrids are purely elastic elements with normal stiffness but no flexural rigidity. They cannot sustain compressive forces and they are connected to the finite element mesh by 3 or 5 nodes, depending on the type of finite element used in the mesh (6 or 15-noded elements). Geogrids are often used to model reinforced earth structures, geotextiles and anchors


GeogridsStructural elements are often combined together to simulate the mechanical behaviour of real engineering structures, such as grouted anchors, which are modelled through a combination of node-to-node anchors and


geogrids.

Interfaces Interfaces are special elements particularly thought for the soil-foundation interaction.

Their effect is a reduction of contact friction, thus enabling a more realistic modelling of the mechanical behaviour than a perfectly glued contact type, which would be what one obtains without introducing any interface. Therefore, interface elements allow relative displacements between structure and subsoil. As usual with special elements, the number of nodes used depends on the


element type used for the soil: 6-nodes 3-integration points interface elements are used in combination with 6-noded finite elements, 10-noded 5-integration points interface elements are used with 15-noded elements.

Interfaces The mechanical behaviour of interface elements is described as function of surrounding elements. An elastic-perfectly plastic constitutive law is assumed, where the strength is obtained from the surrounding soil according to:

inter inter soilc R c=

inter inter soiltan R tan = The user is requested to input the reduction factor Rinter and the strength


parameters of the interface elements are determined from the strength of soil from equations above

Typical values which can be given to the reduction factor are given as follows:Materials interaction: Rinter:Clay/Steel 0.5Sand/Concrete 1.0 - 0.8Sand/Steel 2/3Clay/Concrete 1.0 - 0.7Soil/Geotextile 1.0Soil/Geotextile 0.9 - 0.5

Interfaces Interface elements are used to reduce the high gradients of stress which are observed in proximity of sharp edges of structural elements. As shown in Figure, the stress distribution can be smoothened by extending the interfaces well beyond the edges.


Interfaces Another common use of interfaces is to make retaining walls (modelled by plates) impermeable (consolidation or flow problems)


structural elements and interfaces

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