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geotechnical finite element lecture note
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Dr. Zhenhe (Song) [email protected]
GHD Pty Ltd
Civil Engineering Analysis and Modelling (CIVL3140)
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Part 1 Geomechanics (Plaxis) Dr. Zhenhe Song
Part 2 Hydraulics (Fluent) A/Prof. Tongming Zhou (Unit coordinator)
Part 3 Structures (Multiframe) Mr. Philip Christensen [email protected]
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All the students to set up PLAXIS Version 9 software before tutorial.If you get your laptop this year, you may have PLAXIS 2010, you need to reinstall Plaxis V9Please try to run PLAXIS in your laptop and make sure it works well.Please ask help from the IT support if you have any problems to open PLAXIS. IT Support: Keith Russell [email protected]
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2x2hrs sessions per week
First 2hrs: Lecture (Theory)Second 2hrs: Tutorials (Practice)
4 weeks in total
6% Weekly Practice; 14% Assignment
40% Exam (combined)
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This note has incorporated the note from previous teaching by Prof. Yuxia Hu
The development of tutorial questions by Dr. Long Yu
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Finite element analysis in geotechnical engineering: theory, David M. Potts, Lidija ZdravkoviFinite element analysis in geotechnical engineering: application, David M. Potts, Lidija ZdravkoviGuidelines for the use of advanced numerical analysis, David Potts, Kennet Axelsson, Lars Grande, Helmut Schweiger and Michael Long
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Modelling and FEM in Geotechnical Engineering
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StabilityLoading on StructureMovement
Footing;Retaining Wall and Deep Excavation;Piles and Bridge Abutment;Embankment, Dams and Seawalls;Tunnel;Stockpile;Dynamic (Seismic Analysis)
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Soils are neither elastic, nor homogeneous.
Soils around the world vary.
Same soil with different saturations and consolidations behaves differently.
Soil properties are difficult to measure.
In situ vs laboratory testing …
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13New civil engineer: 14/04/2005
Geotechnical engineering is complex. It is not because you’re using the FEM that it becomes simpler;The quality of a tool is important, yet the quality of a result (mainly) depends on the user’s understanding of both the problem and the tool;The design process involves considerably more than analysis.
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Traditional methods of analysis often use techniques that based on assumptions that over simplify the problem at hand.
These methods lack the ability to account for all of the factors and variables the design engineer faces and may severely limit the accuracy of the solution.
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Equilibrium (stress)
Compatibility (strain)
Constitutive Relationship (stress-strain)
Boundary Condition
Solution of GeotechnicalProblems
Numerical“Exact” or Closed Form
Empirical, Based on Experience
LimitAnalysis
DiscreteElement
FiniteElement
FiniteDifference
BoundaryElement
Finite/BoundaryElement
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LimitEquilibrium
Method of Analysis Solution Requirements Design Information
Stress Equilibrium
Compatibility Constitutive behaviour
Stability Displacements
Limit equilibrium (P) XRigid plastic
X
Slip-line method (P) XRigid plastic
X
Limit Analysis-Lower Bound-Upper Bound X
XPerfectly plastic
XX
Displacement finite element Any
P– partially satisfied
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Receive Design Prescriptions(from a client)
Obtain Soil Properties(Site investigations and lab testing)
Model Geotechnical Problem
Detailed Design Report
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Verification
http://www.cofs.uwa.edu.au/Researh/centrifugeprojects.html
http://www.pbase.com/image/41209293
Geotechnical modelNumerical modelling
Physical modellingSilo
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Plain strain or axisymmetric
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Footing (B/2)
CL
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Discretisation (mesh):Divide the model field (soil and/or structure) into parts (nodes and elements)
Displacement Approximation: Over each part (element), displacement is expressed as function of nodal values
Element Equation: Use an approximate variational principle (e.g. minimum potential energy) to derive an element equation KUE=PE
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Global Equation: Then assemble the parts of element equation to form a global equation KU=P
Boundary Condition: Formulate boundary conditions and modify global equations. Loads affect P, displacement affect U
Solutions: Solve displacement values at nodes and then stress and strain can be evaluated
Footing (B/2)
Elementx
xx
Node
Gauss point (integration point)x
CL
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Element Type Degree of Freedom
per Element
Plane Strain Axisymmetric
Integration rule
Gauss point
Constraints per
Element
Ratio Degrees of Freedom Constraints
Suitable Integration rule
Gauss point
Constraints per
Element
Ratio of Degrees of Freedom Constraints
Suitable
3-noded constant Strain
triangle
1 1-point 1 1 Y 3-point 3 1/3 N
6-noded linearStrain
triangle
4 3-point 3 4/3 Y 6-point 6 2/3 N
10-noded quadraticStrain
triangle
9 6-point 6 3/2 Y 12-point 10 9/10 N
15-noded cubicStrain
triangle
16 12-point 10 8/5 Y 16-point 15 16/15 Y
4-nodedquadrilateral
2 2x2 3 2/3 N 3x3 5 2/5 N
8-noded quadrilateral
6 3x3 6 1 Y 3x3 9 2/3 N
12-nodedquadrilateral
10 4x4 10 1 Y 4x4 13 10/13 N
17-nodedquadrilateral
16 5x5 14 8/7 Y 5x5 19 16/19 N
Sloan, S. W. and Randolph, M. F. (1982) “Numerical prediction of collapse loads using finite element analysis”, Int. J. Num. Ana. Meth. Geo.
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xy
u
v
1 2
3Function:u(x,y) = a1 + a2x + a3yv(x,y) = b1 + b2x + b3y
(x1, y1)u1, v1
(x3, y3)u3, v3
(x2, y2)u2, v2
u1 = u(x1, y1) = a1 + a2x1 + a3y1
u2 = u(x2, y2) = a1 + a2x2 + a3y2
u3 = u(x3, y3) = a1 + a2x3 + a3y3
3
2
1
33
22
11
3
2
1
111
aaa
yxyxyx
uuu
u = ?
Solve for a1, a2, a3
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2A)xy(x)yx(y)yxy(x
2A)xy(x)yx(y)yxy(x
2A)xy(x)yx(y)yxy(x
NNN
N
12211221
31133113
23322332
3
2
1
3
3
2
2
1
1
321
321
N0N0N00N0N0N
vu
U
vuvuvu
Function of (x,y)
Function of (x,y)
6
6
5
5
4
4
3
3
2
2
1
1
654321
654321
N 0 N 0 N 0 N0N0N00 N 0 N 0 N 0N0N0N
vu
U
vuvuvuvuvuvu
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xy
u
v
1 2
3
(x1, y1)u1, v1
(x3, y3)u3, v3
(x2, y2)u2, v2
u = ?
6 5
4
(x6, y6)u6, v6
(x5, y5)u5, v5
(x4, y4)u4, v4
Function:u(x,y) = a1 + a2x + a3y + a4x2 + a5xy + a6y2
v(x,y) = b1 + b2x + b3y + b4x2 + b5xy + b6y2
6
5
4
3
2
1
2666
2666
2555
2555
2444
2444
2333
2333
2222
2222
2111
2111
6
5
4
3
2
1
111111
aaaaaa
yyxxyxyyxxyxyyxxyxyyxxyxyyxxyxyyxxyx
uuuuuu
Strain within an element:Displacement:u(x,y) = a1 + a2x + a3y + a4x2 + a5xy + a6y2
v(x,y) = b1 + b2x + b3y + b4x2 + b5xy + b6y2
Strain:
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u
v
1 2
3
6 5
4yaxaa
xu
542xx 2
ybxbyv
653yy 2b
ybaxbaabxy
yu )2()2()( 564532xy
eUBe
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Constitutive RelationStress and strain can be written in vector form and then expressed as
DLinear isotropic elasticity
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1 2
3
6 5
4P1x
P1Y
Body forces and surface tractions applied to the element may be generalized into a set of forces acting at the nodes
Based on an appropriate variationalprinciple (e.g. minimum potential energy) to derive element equations:
ee PUeKwhere
vDBdBK TeIn order to get [Ke], integration (gaussianintegration) must be performed for each element. Basically, the integral of the function is replaced by weighted sum of the function at a number of integration points
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The stiffness for the complete mesh is evaluated by combining the individual element stiffness matrixes assembly)
This produces a square matrix K of dimension equal to the number of degree-of-freedom in the mesh
The global vector of nodal forces P is obtained in a similar way by assembling the element nodal force vectors
The assembled stiffness matrix and force vector are related by:
PUK
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144
134
133
124
123
122
114
113
112
111
133
143
144
124
123
122
114
113
112
111
KKKKKKKKKK
KKKKKKKKKK
266
256
255
246
245
244
144
236
235
234
134
233
133
124
123
122
114
113
112
111
255
265
266
245
246
244
235
236
234
233
KKKKKKKKKKKKK
KKKKKKK
KKKKKKKKKK
Find symmetrical features, central line can be a roller boundary. (CL) (1)Soil domain needs to be large enough to avoid boundary effect. (10x(B/2), 10x(B/2))The bottom boundary can be fixed boundary. (2)The side boundary can be roller boundary. (3)Top boundary is normally a free boundary. (4)
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CL
Footing (B/2)
10x(B/2)
10x(B/2)1
2
3
4
Element size: the smaller, the more accurate
Element type: the higher order, the more accurate
Boundary conditions: domain size, realistic
Constitutive model: complexity economy
Soil parameters: realistic, measurable
Understanding of the real problem numerical
model
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Less elements to reduce computation timeSmaller elements to increase accuracy
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Optimum MeshCombination of coarse and fine mesh
How ?
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Footing (B/2)
Displacement control (prescribed displacement) or
load control (prescribed load) ?
2-dimensional or 3-dimensional analysis ?
Plain strain or axisymmetric ?
Drained, undrained or consolidation analysis?
Construction Stages
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Pre-processingDefine problem(2D or 3D? Plain strain or Axisymmetric? Soil model? Drained or undrained?); define domain (size?); define boundary condition; generate mesh (element type? mesh density?); input soil/foundation parameters (worked out soil parameter from site investigation).
2) CalculationFEM Calculation Steps
3) Post-processingProcess calculation results, such as soil stress/strain distribution; soil deformations, et al.
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