MR14

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STRUCTURAL AND GEOTECHNICAL ENGINEERING DEPARTMENT

ROCK MECHANICS 2ROCK MECHANICS 2

Giovanni Barla

Politecnico di Torino

LECTURE 14 - OUTLINEThe Finite Element Method2D and 3D Problems - Nonlinear materials

(a) Introduction(b) Solution procedures

- Iterative method(c) Examples

FEM FOR NONLINEAR MATERIALSFEM FOR NONLINEAR MATERIALS

(a) Geometrically Nonlinear Problems

P(a1) Geometrically linear

(a2) Geometrically nonlinear

P

In geometrically nonlinear problems, the final deformed structure is significantly different from the initial undeformed structure. In such a case the strain-displacement relation is to consider higher order terms,e.g.:

∂u∂x + 1

2 ( ((∂u∂x )2∂w

∂x∂v∂x )2)2 + + etc.

1 2 3 4 5h h

h

h

15

2

3

4

DeformedStructure

Initial Geometry

εx=

H H

A B C

H1 2 3

Fi

L

ci

Yi

Nonlinear constitutive lawfor element i (i=1,2,3)

yield

(a) Nonlinearity introduced by the constitutive behaviour

A problem can be nonlinear from the so-called physical point of view. In such a case the nonlinearity is due to the nonlinear constitutive behaviour.


2

0.00

10.0

20.0

30.0

40.0

50.0

60.0

0.0 10.0 20.0 30.0 40.0 50.0 60.0

Max. disp = 0.12 m

DESIGN ANALYSESDESIGN ANALYSES

Comparison of closedComparison of closedform solution form solution (CFS) (CFS) andandFDMFDM results results CFS FDM CFS FDM ur ur (cm) 14.0 12.0 (cm) 14.0 12.0Rpl Rpl (m) 11.8 12.4(m) 11.8 12.4

NOTENOTEThe same problem asThe same problem asabove for above for a a circular circular tunneltunnel((diameterdiameter 11.5 m) 11.5 m) andandrock mass rock mass properties asproperties asdefined for the characteristicdefined for the characteristicline calculationline calculation

Circular Tunnel Underground Cavity

In both cases the rock mass around the opening is givenan elasto-plastic behaviour


The nonlinearity of the stress [σ] strain [ε] law causes the governingfinite element equations

[K]G [u]G = [R]G

to be reduced to the following incremental form:

[K]iG [Δu]i

G = [ΔR]iG

where:

[K]iG is the incremental global system stiffness matrix

[Δu]iG is the vector of incremental nodal displacements

[ΔR]iG is the vector of incremental nodal forces

i is the increment number continue


Application of FEM nonlinear theory to the uniaxial loading of a barof nonlinear material

ε

Eo Ko

Applied load, R

σi-1

σi

σ

Ei

Ei-1 Ki-1

Ki

R

uεi-1 εi ui-1 ui

[R][u][K]

[σ][ε][C]

DIRECT ITERATIVE APPROACH (this approach is the first one used in FEM modelling of nonlinear problems in continuum mechanics)

DIAGRAMMATIC REPRESENTATIONOF THE DIRECT ITERATIVE APPROACH

R

u

Applied Load Rap

Ko

Δu1 Δu2

R1

R2

K1

• With the first iteration (i=1)

[u]1 = [K]0-1 [R]ap (14.1)

u1 u2

• With the second iteration (i=2)

1) Evaluate the stiffness element per element in the system, based upon the state of stress-strain obtained at the end of iteration 1. Therefore, we can write the system stiffness

matrix [K]12) Compute:

[u]2 = [K]1-1 [R]ap (14.2)


continue

3

• With the i-th iteration (i)

[u]i= [K]i-1-1 [R]ap (14.3)

The iterative approach is applied up to obtaining in the(i+1) iteration the ui+1 displacement , not very far from theui displacement computed in the i iteration:

ui+1- ui ≈ 0,

conversely the “non equilibrated load” (Rap - Ri)for iteration i+1 is very small,

(Rap - Ri) ≈ 0

It follows from equation (14.3) that we need to compute the system stiffnessmatrix for each iterative step.

continue

EXAMPLE (Direct Iterative Approach)da Zienkiewicz e Cheung (1967) (*))

EXAMPLE (Direct Iterative Approach)da Zienkiewicz e Cheung (1967) (*))

σ

ε

ε =23

√ (ε1- ε2)2 + (ε2- ε3)2 + (ε3- ε1)2

σ = 13

(σ1- σ2)2 + (σ2- σ3)2 + (σ3-σ1)2 √

ε0

E and ν will be some complicated butknown functions of the stress σ or strainε components -> [C]s

• The iterative approach is described bythe following steps:

1) The full load is applied to the structure andthe elastic stresses and strains determinedbased on E and ν values corresponding tozero stress (initial parameters E0 e ν0 give [C]s )

Es

2) New values of E and ν (i.e. [C]s) are determined for each element, depending on the state of stress (or the strain) reached in the previous step

3) An elastic analysis is again carried out, based on the values of elastic constantsfound for each element in 2) above

(*) first applications of FEM to the solution of nonlinear problems

4) Step 2) is repeated

5) Step 3) is repeated, etc.

Rap= load appliedto the stucture

continue

R

A

cu

Ei

Ep

A

B

C

Problem: The elements shown are given the stress - strain law ABC (see Figure)(Ei = 25 GPa, Ep = 2.5 GPa). The transition from the linear elastic (ILE) to the nonlinearelastic behaviour (hardening) is defined by the Tresca’s criterion τmax = cu (cu = 100 kPa)

τn

σn

σ

ε

AB

BC

u=v=0

continue

EXAMPLE: Uniaxial loading of a specimenEXAMPLE: Uniaxial loading of a specimen EXAMPLE: Uniaxial loading of a specimenEXAMPLE: Uniaxial loading of a specimen

1 2 3 4 5 6 7 8 9 10

0.2

0.4

0.6

0.8

1.0σy(MPa)

Number of iteration

ε

σ

εA εB

B

A A’

C

Iterative Approach Change of the σy stressversus the iteration number

The approach is convergent tothe required solution

continue

4

• A possible application of the direct iterative approach regards the analysis of soil -structure interaction problems as in the case of the soil - lining interaction analysisof tunnels. The method is known as the “ bedding modulus method” or the “hyperstatic reaction method”

• The lining is discretized by using either “beam” elements or “plane”elements. The ground reaction forces, which result from structure deformations, are simulated with the aid of bedding moduli or spring bars which act at the nodal points

x xx

yyy

1

2

1

2ground

lining

ph

pv

Method Method LST

interaction

loads

SOIL - STRUCTURE INTERACTIONFOR THE ANALYSIS OF A TUNNEL LINING


The method requires the following:Definition of pressures of soil or rock masses and water, which have

an effect on the structure, as well as single loads have to be pre -determined as external loads

The ground - reaction forces, which result from deformations, are computed and the state of stress in the lining is calculated accordingly

Interaction Model (spring bars)

[K][u]

[P]

[K]

On the contour



example example





example example

Moment Normal Force

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ANALISI DI INTERAZIONE CON IL TERRENOVERIFICHE DEL RIVESTIMENTO DI UNA GALLERIA

ANALISI DI INTERAZIONE CON IL TERRENOVERIFICHE DEL RIVESTIMENTO DI UNA GALLERIA

example example

Stress Reinforcement

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