APPENDIX   A

TRANSFORMATION MATRICES

A.1  Transformation of Coordinates

A.2  Transformation of Displacements

A.3  Transformation of Strains

A.4  Transfomation Stresses

A.5  Transformation due to Rotation of Axis

A.6  Transformation of Two-Dimensional Case

A.7  Transformation of Elastic Constants and Compliances

A.1 Transformation of Coordinates

            The coordinates are vectors. Hence the rules associated with transformation of vectors can be used for transformation of coordinates. Figure A.1illustrates two mutually perpendicular coordinate system x₁ x₂ x₃ and x_1' x_2' x_3' (not necessarily Cartesian coordinates) oriented with respect to each other such that

                                                                                                 (A.1)

where

                                                                                            (A.2)

is the rotation matrix and its elements represent the direction cosines of angles between axis systems x₁ x₂ x₃ and x_1' x_2' x_3'. These are listed below:

                                                                     (A.3)

Fig. A.1

Note that rotational angles  are considered positive when measured from the x₁ x₂ x₃ system to x_1' x_2' x_3' system. The rotational matrix [T_r] is orthogonal i.e., [T_r]^-1. Thus, when a transformation is sought from the x_1' x_2' x_3' coordinates to the x₁x₂x₃ coordinated, one can write

                                             (A.4)

A.2 Transformation of Displacements

            Consider that u₁u₂ and u₃ are displacement components with respect to the coordinate system x₁ x₂ x₃  and  are those corresponding to the x_1' x_2' x_3' system. As displacements are also vectors, similar to coordinates, one can write

                             (A.5)

            The rotation matrix [T_r] can also be used for coordinate transformation of other vectors such as rotational displacements, forces and moments.

A.3 Transformation of Strains

            Here it is intended to relate strain components  corresponding to the x_1' x_2' x_3' coordinates to strain components  corresponding to the x₁x₂x₃ coordinate system.

            Now,  Using  chain rule of differentiation,

                                                                       (A.6a)

with .

In a similar way,

                                                              (A.6b)

                                                                   (A.6c)

Noting from Eq. A.2 that

and substituting Eqs/ (A.6) in it, we obtain

or

Proceeding in a similar way, it can be shown that

or,                                                                              (A.7)

where,  

                                                                           (A.8)

with

Conversely,                                                                            (A.9)

A.4 Transfomation Stresses

            For stress transformation we relate stress components  in the x₁x₂x₃ coordinates to stress components  in the x_1' x_2' x_3' coordinates.

            Let  be virtual strain components in two coordinate systems. The work done by stresses due to virtual displacements does not change when computed in two coordinate systems. Equating the work computed in two coordinate systems.

i.e .,                                                                                      (A.10)

Conbersely,                                                         (A.11)

Note that

                                                                                                   (A.12a)

It follows that

                                                                                                 (A.12b)

                                                                                                   (A.12c)

To determine , we know from Eq. (A.8) that

Then,

                                                                        (A.13)

A.5 Transformation due to Rotation of Axis

            Consider the case of a simple inplane rotation  about the x₃ axis (the axis x'₃ is assumed to coincide with the x₃ axis as shown in Fig. A.2. The rotation matrix [T_r] is then reduced to

                                                                                                    (A.14)

with m = cos  and n = sin

Fig. A.2

The stress and strain transformation matrices  and   then take the following forms:

                                                                          (A.15)

and

                                                           (A.16)

A.6 Transformation of Two-Dimensional Case

            If transformation is required from the two-dimensional x₁x₂ coordinate system to the x_1' x_2'  system only (Fig. A.3), the rotation matrix  further simplifies to

                                                                                                         (A.17)

with m = cos  and n = sin

Fig. A.3

Then

                                                                                (A.18)

                                                                       (A.19)

A.7 Transformation of Elastic Constants and Compliances

            The stress-strain relations are expressed as

             in the x₁x₂x₃ coordinates                                                 (A.20)

and        in the x_1' x_2' x_3' coordinates                                           (A.21)

Here we express [C] in terms of [C'].

From Eq. A.10, we have

or                                                             [From Eq. A.21]

                                                                 [From Eq. A. 7]

or          

with                                                                                  (A.22)

Proceeding in a similar manner, but using stress strain relations in terms of compliances as follows

                                                                                                   (A.23)

and       ,

it can be shown that

                                                                                               (A.24)