Download presentation

Presentation is loading. Please wait.

Published byLizette Peek Modified about 1 year ago

1
“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, 12-15 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
Dilatation (change of area) at a point P: beforeafter (relative change of area) Dilatation may be positive (expansion) or negative (shrinkage) 1 Three types of relevant local deformation parameters dilatation

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

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

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

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

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

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

12
beforeafter Planar deformation is completely determined by the gradient matrix F

13
Diagonalization of F beforeafter Planar deformation is completely determined by the gradient matrix F

14
Diagonalization of F beforeafter Planar deformation is completely determined by the gradient matrix F

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

16
Diagonalization of F beforeafter Planar deformation is completely determined by the gradient matrix F

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

18
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 F q = I : simple but inappropriate because it refers to oblique axes and non unit basis vectors

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

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

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

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

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

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 k 1 and its minimum value k 2 (principal curvatures)

26
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 k 1 and its minimum value k 2 (principal curvatures) 90 Deformation of a curved surface – Bending

27
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 k 1 and its minimum value k 2 (principal curvatures) The curvature of any normal section at angle from the first principal direction is given by Deformation of a curved surface – Bending

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

29
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 Deformation of a curved surface – Bending

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
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. Deformation of a curved surface - Interpolation

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google