“Measuring the Changes” Joint Symposium 13th FIG Symposium on Deformation Measurement and Analysis 4th IAG Symposium on Geodesy for Geotechnical and Structural.

“Measuring the Changes” Joint Symposium13th FIG Symposium on Deformation Measurement and Analysis

4th IAG Symposium on Geodesy for Geotechnical and Structural EngineeringLisbon, Portugal, 12-15 May 2008

Derivation of Engineering-relevant Deformation Parametersfrom Repeated Surveys of Surface-like Constructions

Athanasios Dermanis

Department of Geodesy and SurveyingAristotle University of Thessaloniki

Surface-like Construction:

One dimension insignificant compared to the other two.

Behaves like a thin-shell

Usual surveying approach to “deformation”:

Determination of 3-dimensional displacements

between two epochs t and t (usually at particular discrete points)

The construction engineer approach to “deformation” (Strength of material point of view):

Local deformation (strain) related to forces (stresses)through the constitutive equations of the material(stress-strain relations)

Dilatation (change of area) at a point P:

ttbefore after

P PEE

0limE

E E

E

(relative change of area)

Dilatation may be positive (expansion) or negative (shrinkage)

1

Three types of relevant local deformation parameters

dilatation

positive dilatation (expansion)

1

Relates to central stresses (forces)

ttbefore after

dilatation



negative dilatation (shrinking)

1

Relates to central stresses (forces)


ttbefore after

dilatation


Shear strain at a point P in a particular direction:

P P

tan

We seek the direction where maximum shear occurs

2

directionof shear

before after

shear strain


ttbefore after

Maxumum shear strain at a point P:

P

Relates to shearing stresses (tearing forces)

2


ttbefore after

shear strain

Bending at a point P in a particular direction: Realized by the change of the radius of curvature R of the corresponding normal section

P P

We seek the direction where maximum bending R-R occurs

R R

(radius of curvature = radius of best fitting circle)

3

bending


ttbefore after

P P

R R3

bending


ttbefore after

Bending at a point P in a particular direction:

Relares to bending torques

x x

x y

y y

x y

F x

yy

x

t tbeforeafter

Planar deformation is completely determined by the gradient matrix F

x

yy

x

1

2

0

0

cos sin

sin c

cos sin

sio n coss

F

Diagonalization of F

t tbeforeafter


x x

x y

y y

x y

F

x

yy

x

1

2

0

0

cos sin

sin c

cos sin

sio n coss

F

1

2

0

0

1

1

12

x

yy

x


t tbeforeafter


( )R( )R

x x

x y

y y

x y

F

x

yy

x

1

2

0

0

1

1

1

x

yy

x


t tbeforeafter


( )R( )R

Deformation only from the eigenvalues of F !

x x

x y

y y

x y

F

2

x x

x y

y y

x y

F x

yy

x

1

2

0

0

1

1

1

x

yy

x


t tbeforeafter


( )R( )R

1 2 1 1 2

1 2

2

P

For dilatation and maximum shearthe surface is approximated by thelocal tangent plane at P(best fitting plane)

Deformation of a curved surface

P

Deformation of a curved surface

For dilatation and maximum shearthe surface is approximated by thelocal tangent plane at P(best fitting plane)

For maximum bendingthe surface is approximated by thelocal oscillating ellipsoid at P(best fitting ellipsoid)

Deformation of a curved surface – Dilatation and maximum shear strain

P

X

Y

Z

PX

PY

On a curved surface onlycurvilinear coordinates (u,v) can be used

Coice of curvilinear coordinates:horizontal cartesian coordinates u = X, v = Y at the original epoch t

At second epoch t :Use as coordinates thoseof original epoch t u = X, v = Y

(convected coordinates)

1 0

0 1q

u u

u vv v

u v

F ICurvilinear deformation gradient Fq = I : simple but inappropriatebecause it refers to oblique axesand non unit basis vectors

u

v

u u

original epoch t second epoch t

v v

q F I


x

y

x

y

u

v

u u


1S 1S

v v

q F I

transformation toorthonormal systems


x

y

x

y

u

v

u u


1S

v v

q F I


S

x

y

x

y

u

v

u u


1S

1 1q

F S F S S S

v v

q F I


S

1

2

0cos sin cos sin( ) ( )

0sin cos sin cos

F R LRDiagonalization

, 90

1 2

1 2

1 2 principal elongations

1 2 1

1 2arctan2

dilatation

maximum shear strain at direction angle

Invariant (independent of coordinate systems) deformation parameters

directions of principal elongations , respectively1 2,


Deformation of a curved surface – Bending

P

Normal section at P:intersection of surface with any planecontaining the surface normal at P

R = radius of circle best fitting to normal section

k = 1/R curvature of normal section at P

R

Among all normal sections there are two perpendicular principal directionswhere the curvature obtains its maximum value k1 and its minimum value k2

(principal curvatures)

P






90


P






The curvature of any normal section at angle from the first principal directionis given by

2k1k( )k

2 21 2( ) cos sink k k


u

u u

u

X

Yu

Z

xx

v

v v

v

X

Yv

Z

xx

T Tu u u vT Tu v v v

x x x xG

x x x x

2 2

2

2 2

2

T T

T T

u u v

u v v

x xn n

Lx x

n n

| |u v

u v

x x

nx x

22 11 12 12 11 222

2det

G L G L G LH

G

det

detK

L

G

22k H H K 2

1k H H K

First Fundamental Form Sencond Fundamental Form

Tangent vectors Normal vector

Mean curvature Gaussian curvature

Computation of principal curvatures


P

2 21 2( ) cos sink k k


P ( )

2 21 2( ) cos ( ) sin ( )k k k

( ) ( ) ( ) maxk k k

Value for maximum from numerical solution of a non-linear equation̂

1ˆ ˆ( )ˆ

k kR

1ˆ ˆ( )ˆ

k kR

Most differing radii of curvature over all normal sections through P

2k1k( )k

2k1k( )k


Deformation of a curved surface - Interpolation

To compute dilatation, maximum shear strain and maximum bending we needthe following functions

( , )

( , )

( , )

X u v

Y u v

Z u v

( , )

( , )

( , )

X u v

Y u v

Z u v

( , )

( , )

u u v

v u v

Using convective coordinates

the required functions reduce to

( , )Z Y

X

Y Y

X

X

(( , )

( , )

( ,

, )

( , )

( , ))

X X Y X

Y X Y Y

X X

Z X Y

Y

Y X Y

Z YZ X

, , ,u X v Y u u X v v Y

They can be obtained from the interpolation of the available discrete data

( , )i i i iX X Y X X

( , )i i iZ Z X Y

( , )i i i iY X Y Y Y ( , )i i i iZ X Y Z Z

Interpolation of the available discrete data

( , )i iX X Y( , )i iZ X Y ( , )i iY X Y ( , )i iZ X Y

In general:Interpolate a function from discrete data( , )f X Y ( , )i i if f X Y

One possibility: Collocation (Minimum norm interpolation = Minimum Mean Square Error Prediction)

Use two-point covariance function: ( , ; , )C X Y X Y

1( , ) Tf X Y c C f

( , )i i if f X Y ( , ; , )ik i i k kC C X Y X Y ( , ; , )i i ic C X Y X Y

1( , ) ,Tf

X YX X

cC f

2 21( , ) ,

TfX Y

X Y X Y

cC f etc.

Deformation of a curved surface - Interpolation

“Measuring the Changes” Joint Symposium 13th FIG Symposium on Deformation Measurement and Analysis 4th IAG Symposium on Geodesy for Geotechnical and Structural.

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