Coordinate Transformation

Foundations of Finite Element Methods WS 2014/15 Coordinate transformation transformation

y

F 2y F 2y F 2x

F 1y α

F 1y

F 1x

cos

cos

F 2x

sin sin

2

1

F 1y

α

x

F 1x

y

F 1y

F 1x

sin

F 1y

F 1y 1

x

α cos

F 1x

cos

1

sin

F 1x

F 1x α

Figure 1: Node forces in the local (element) and global coordinate system

Transformation of the nodal forces at node 1: F 1x  =  F 1x cos(α) −  F 1y sin(α)

F 1x  =  F 1x cos(α) +  F 1y sin(α)

F 1y =  F 1x sin(α) +  F 1y cos(α)

F 1y =  − F 1x sin(α) +  F 1y cos(α)

In the same way, way, the forces can be transformed at node 2. Thus, Thus, the transformation transformation of the forces from the   global coordinate coordinate system   to the   local local coordinate coordinate system  can be written in the matrix form F 1x F 1x cos(α) sin(α) 0 0 0 0 F 1y F 1y − sin(α) cos(α) = 0 0 cos(α) sin(α) F 2x F 2x F 2y 0 0 − sin(α) cos(α) F 2y

  

  

     

  

or using symbolic notation

� � = [ F 

  

L] {F } .

We deriv derived ed this expressi expression on for one elemen element. t. In line with the lectur lecturee notes notes (cp. 1.1 1.13), 3), we can write for the vector of generalized forces r e

= Le r e .

Using the equation (1.16) from the lecture notes e

K

= Le T Ke Le ,

we can now transform the element stiffness matrix Ke w.r.t. the local system to the element stiffness matrix Ke w.r.t. the global coordinate system. This yields

K

e

  =  k  

cos2 α cos α sin α − cos2 α − cos α sin α 2 cos α sin α sin α − cos α sin α − sin2 α cos2 α cos α sin α − cos2 α − cos α sin α 2 cos α sin α sin2 α − cos α sin α − sin α

  , 

where k  denotes  denotes the stiffne stiffness ss of the truss. truss. This This matrix matrix must be calcul calculate ated d for each truss truss in a truss-system. Afterwards, all element stiffness matrices can be assembled to the global stiffness matrix. 1