How to Work with Curved Structures?

Transcription

1 How to Work with Curved Structures? Juha Paavola & Eero-Matti Salonen New Trends and Challenges in Civil Engineering Education Patras, Greece November 2011

2 Contents Background Why do we need improvements? What have we done? Basic tools for modeling Some simple applications

3 Background There is a strong need to develop educational processes of theoretical topics Computational visualization, ideas of modern pedagogy and other strategies are the most often applied improvements - the mechanics or the theories themselves have got only the minor role We have tried to find out some tools to help students to become inspired in the theoretical matter, which is based mathematically on rudiments, only (vector calculus)

4 Why have we done what we have done? There are basically two main reasons:

5 Problems of Mechanics Incoherence Beam theories Plate theories Shell theories Stability Numerics Dynamics Nonlinear problems

6 and secondly The figures of differential geometry

7

8 Minor challenges Teaching is traditionally too methodoriented Mechanics has a strong overlapping with Mathematics The fear of curved geometries

9 What is then needed? General common mathematical tool, which is based on rudiments only applicable to all kind of problems exact with no dubioucity easy to learn

10 The idea for improvement INPUT Geometry Kinematics Mathematical manipulator or Mill OUTPUT Equations needed, strains, equilibrium etc.

11 How will this be done? The medicine we are serving Vector calculus defining the geometry and kinematics in vector fields Use of a local Cartesian frame a way to avoid defining various derivatives in curvilinear coordinates Energy principles and principle of virtual work

12 Geometry description is given in curvilinear co-ordinates by using the position vector r = r { α, β, γ}

13 Position vector

14 Geometry is given in curvilinear co-ordinates by using the position vector mathematically it defines the domain to be considered it defines also all the mathematical operators

15 The role of the local frame according to B. Irons I e α I = e O α J = e β α J β e β The unit vectors of local frame are constant, both in magnitude and direction

16 Kinematics Kinematics is a tool controlled by the analyst himself both in analytical and numerical analyses It can be interpreted as the freedom of the structure to be allowed to deform given by the user It will define whether the problem is 1-, 2- or 3- dimendional

17 1-, 2- or 3-dimensional v ( x,y) v ( x,y,z) x x y v ( x) z x y

18 Local Cartesian frame Strains are defined in local frame linear ones ε u u u = e, γ = e + e, etc. X X Y X X XY Y X or non-linear ε γ X u 1 u u = e X +, X 2 X X u u u u = e + e +, etc. X Y X Y XY Y X

19 This image cannot currently be displayed. While the kinematics is given in curvilinear coordinates, the chain rule for differentiation is needed X Y Z = + + α α X α Y α Z X Y Z = + + β β X β Y β Z X Y Z = + + γ γ X γ Y γ Z 1 r r r X I J K α α α α r r r = Y I J K β β β β r r r Z I J K γ γ γ γ In orthogonal systems the transformation matrix will be diagonal

20 The kinematics is given in curvilinear geometry u( α, β, γ ) = ue + ve + we α β γ and the derivatives are calculated simply u u eα v eβ w eγ = eα + u + eβ + v + eγ + w α α α α α α α All the troubles due to curvilinearity are included in these

21 Principle of virtual work Replaces also the use of the figures of differential geometry Represents as an exact mathematical formulation Is usable equally well in complicated geometries and non-linear analyses

22 Some Simple Applications

23 Geometry model r r e = e s X r e y y = r e Y s R r r r r = ( R+ y) e y + ze z

24 Kinematics Extension, Bending and Shear u ( s,y,z) = ( u yθ zθ ) e + ( v zθ ) e + ( w + yθ ) is simplified to z y s s y s z ( ) ( ) u s,y = u yθ e + ve z s y e with u= u( s), v= v( s), θ = θ ( s), e = e ( s), e = e ( s) z z s s y y

25 Kinematics Bending, Torsion and Shear ( s,y,z) ( u yθ zθ ) + ( v zθ ) + ( w + yθ ) u e e e = z y s s y s z u e e e is simplified to ( s, y,z) = zθ zθ + ( w + yθ ) y s s y s z w= w( s), θ = θ ( s), θ = θ ( s), e = e ( s), e = e ( s) s s y y s s y y

26 Kinematics Two-dimensional inplane-bending u( s,y)= ue + ve s with y u= u( s, y), v= v( s, y) and e = e ( s, y), e = e ( s, y) s s y y

27 Kinematics Two-dimensional bending u( s, y, z) = u( s, y) zθ y( s, y) es( s, y) [ θ ] + v( s, y) z ( s, y) e ( s, y) + w( s, y) e s y z

28 Thank you for your time!