1. Derivation of Engineering-relevant Deformation Parameters
“Measuring the Changes” Joint Symposium
13th FIG Symposium on Deformation Measurement and Analysis
4th IAG Symposium on Geodesy for Geotechnical and Structural Engineering
Lisbon, Portugal, May 2008
Derivation of Engineering-relevant Deformation Parameters
from Repeated Surveys of Surface-like Constructions
Athanasios Dermanis
Department of Geodesy and Surveying
Aristotle University of Thessaloniki

2. Surface-like Construction:
One dimension insignificant
compared to the other two.
Behaves like a thin-shell

3. Usual surveying approach to “deformation”:
Determination of 3-dimensional displacements
between two epochs t and t
(usually at particular discrete points)

4. 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)

5. 1 Three types of relevant local deformation parameters dilatation
before
after
dilatation 1
Dilatation (change of area) at a point P:
(relative change of area)
Dilatation may be positive (expansion) or negative (shrinkage)

6. 1 Relates to central stresses (forces)
Three types of relevant local deformation parameters
before
after
dilatation 1
Dilatation (change of area) at a point P:
Relates to central stresses (forces)
positive dilatation (expansion)

7. 1 Relates to central stresses (forces)
Three types of relevant local deformation parameters
before
after
dilatation 1
Dilatation (change of area) at a point P:
Relates to central stresses (forces)
negative dilatation (shrinking)

8. 2 Three types of relevant local deformation parameters shear strain
before
after
shear strain 2
Shear strain at a point P in a particular direction:
before
after
direction
of shear
We seek the direction where maximum shear occurs

9. 2 Relates to shearing stresses (tearing forces)
Three types of relevant local deformation parameters
before
after
shear strain 2
Maxumum shear strain at a point P:
Relates to shearing stresses (tearing forces)

10. 3 Three types of relevant local deformation parameters bending
before
after
bending 3
Bending at a point P in a particular direction:
Realized by the change of the radius of curvature R
of the corresponding normal section
(radius of curvature = radius of best fitting circle)
We seek the direction where maximum bending R-R occurs

11. 3 Relares to bending torques
Three types of relevant local deformation parameters
before
after
bending 3
Bending at a point P in a particular direction:
Relares to bending torques

12. Planar deformation is completely determined by the gradient matrix F
after
before

13. Planar deformation is completely determined by the gradient matrix F
after
before
Diagonalization of F

14. Planar deformation is completely determined by the gradient matrix F
after
before
Diagonalization of F

15. Planar deformation is completely determined by the gradient matrix F
after
before
Diagonalization of F
Deformation only from the eigenvalues of F !

16. Planar deformation is completely determined by the gradient matrix F
after
before
Diagonalization of F

17. Deformation of a curved surface
For dilatation and maximum shear
the surface is approximated by the
local tangent plane at P
(best fitting plane)

18. Deformation of a curved surface
For dilatation and maximum shear
the surface is approximated by the
local tangent plane at P
(best fitting plane)
For maximum bending
the surface is approximated by the
local oscillating ellipsoid at P
(best fitting ellipsoid)

19. Deformation of a curved surface – Dilatation and maximum shear strain
On a curved surface only
curvilinear 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 those
of original epoch t
u = X, v = Y
(convected coordinates)
Curvilinear deformation gradient
Fq = I : simple but inappropriate
because it refers to oblique axes
and non unit basis vectors

20. Deformation of a curved surface – Dilatation and maximum shear strain
original epoch t
second epoch t

21. Deformation of a curved surface – Dilatation and maximum shear strain
original epoch t
second epoch t
transformation to
orthonormal systems

22. Deformation of a curved surface – Dilatation and maximum shear strain
original epoch t
second epoch t

23. Deformation of a curved surface – Dilatation and maximum shear strain
original epoch t
second epoch t

24. Deformation of a curved surface – Dilatation and maximum shear strain
Diagonalization
Invariant (independent of coordinate systems) deformation parameters
principal elongations
directions of principal elongations , respectively
dilatation
maximum shear strain at direction angle

25. Deformation of a curved surface – Bending
Normal section at P:
intersection of surface with any plane
containing 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 directions
where the curvature obtains its maximum value k1 and its minimum value k2
(principal curvatures)

26. Deformation of a curved surface – Bending
Normal section at P:
intersection of surface with any plane
containing the surface normal at P
90
R = radius of circle best fitting to normal section
k = 1/R curvature of normal section at P
Among all normal sections there are two perpendicular principal directions
where the curvature obtains its maximum value k1 and its minimum value k2
(principal curvatures)

27. Deformation of a curved surface – Bending
Normal section at P:
intersection of surface with any plane
containing the surface normal at P
R = radius of circle best fitting to normal section
k = 1/R curvature of normal section at P
Among all normal sections there are two perpendicular principal directions
where the curvature obtains its maximum value k1 and its minimum value k2
(principal curvatures)
The curvature of any normal section at angle  from the first principal direction
is given by

28. Deformation of a curved surface – Bending
Computation of principal curvatures
Tangent vectors
Normal vector
First Fundamental Form
Sencond Fundamental Form
Mean curvature
Gaussian curvature

29. Deformation of a curved surface – Bending
original epoch t
second epoch t
Value for maximum from numerical solution of a non-linear equation
Most differing radii of curvature over all normal sections through P

30. Deformation of a curved surface - Interpolation
To compute dilatation, maximum shear strain and maximum bending we need
the following functions
Using convective coordinates
the required functions reduce to
They can be obtained from the interpolation of the available discrete data

31. Deformation of a curved surface - Interpolation
Interpolation of the available discrete data
In general:
Interpolate a function from discrete data
One possibility: Collocation
(Minimum norm interpolation = Minimum Mean Square Error Prediction)
Use two-point covariance function:
etc.