# Summary of Element Transformation Matrices

Shared by:
Categories
Tags
-
Stats
views:
17
posted:
6/7/2009
language:
English
pages:
1
Document Sample

```							CEE 371: Modeling of Structural Systems                                                                   Spring 2004

SUMMARY OF ELEMENT TRANSFORMATION MATRICES
FOR THE STIFFNESS METHOD

Element transformation matrices, [Γ], appear in the following relationships:

Element nodal force transformation:  F       F 

Element nodal displacement transformation:          

Element stiffness matrix transformation:  k       k    
T

In these equations, primed quantities are in element (local) coordinates, while unprimed
quantities are in global (structure, overall) coordinates. The transformation matrices for
1
perpendicular coordinate are orthogonal, that is,        regardless of whether [Γ] is
T

square or rectangular.

For a frame element at orientation  (measured counterclockwise from the x axis to the x'
axis) the specialized forms of [Γ] for various elements studied in this course are:

   cos        sin        0           0 
2-DOF truss element:                                                   
2 4  0             0       cos        sin  

 cos        sin           0            0
                                          
     sin       cos           0         0

4-DOF truss element:           
44  0                  0     cos         sin  
                                          
 0               0      sin       cos  

4-DOF beam element: Not applicable (or [I]) because beams have co-linear x and x'.

 cos                      sin      0          0        0     0
                                                                 
 sin                    cos      0         0         0     0
                                                                 
   0                            0       1          0        0     0
6-DOF beam-column element:                                                                       
66  0                             0       0       cos      sin    0
 0                           0       0        sin    cos    0
                                                                 
 0
                             0       0          0        0     1

Note that each of these transformation matrices for straight two-node elements can be
partitioned and written in the general form:
    0
              
  0   

in which [γ] is the orthogonal nodal transformation for a single node and [0] is a null matrix.

```
Related docs