Summary of Element Transformation Matrices

							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.