Transformation of Field Variables between Cartesian, Cylindrical, and Spherical Components

APPENDIX

B

Transformation of Field Variables between Cartesian, Cylindrical, and Spherical Components

This appendix contains some three-dimensional transformation relations between displacement and stress components in Cartesian, cylindrical, and spherical coordinates. The coordinate systems are shown in Figure A.1 in Appendix A, and the related stress components are illustrated in Figure B.1. These results follow from the general transformation laws (1.5.1) and (3.3.3). Note that the stress results can also be applied for strain transformation.

z

z

τ Rθ

σz

σR τR φ

τθ z

τ rz

σθ τ rθ

φ

σr

σθ

τ φθ y

r x

θ

σφ

y

θ

dθ dr

R

x

(Cylindrical System)

(Spherical System)

FIGURE B.1 Stress Components in Cylindrical and Spherical Coordinates.

537

538

APPENDIX B Transformation of Field Variables

Cylindrical components from Cartesian The transformation matrix for this case is given by 2

cosq

sinq

6 ½Q ¼ 6 4 sinq 0

0

3

cosq

7 07 5

0

1

(B.1)

Displacement transformation ur ¼ u cosq þ v sinq uq ¼ u sinq þ v cosq

(B.2)

uz ¼ w

Stress transformation sr ¼ sx cos2 q þ sy sin2 q þ 2sxy sinq cosq sq ¼ sx sin2 q þ sy cos2 q  2sxy sinq cosq sz ¼ sz

  srq ¼ sx sinq cosq þ sy sinq cosq þ sxy cos2 q  sin2 q

(B.3)

sqz ¼ syz cosq  szx sinq szr ¼ syz sinq þ szx cosq

Spherical components from cylindrical The transformation matrix from cylindrical to spherical coordinates is given by 2

sinf

6 ½Q ¼ 6 4 cosf 0

0 cosf

3

7 0 sinf 7 5 1 0

(B.4)

APPENDIX B Transformation of Field Variables

539

Displacement transformation uR ¼ ur sinf þ uz cosf uf ¼ ur cosf  uz sinf

(B.5)

uq ¼ uq

Stress transformation sR ¼ sr sin2 f þ sz cos2 f þ 2srz sinf cosf sf ¼ sr cos2 f þ sz sin2 f  2srz sinf cosf sq ¼ sq

  sRf ¼ ðsr  sz Þsinf cosf  srz sin2 f  cos2 f

(B.6)

sfq ¼ srq cosf  sqz sinf sqR ¼ srq sinf þ sqz cosf

Spherical components from Cartesian The transformation matrix from Cartesian to spherical coordinates can be obtained by combining the previous transformations given by (B.1) and (B.4). Tracing back through tensor transformation theory, this is accomplished by the simple matrix multiplication 2

sinf

6 ½Q ¼ 4 cosf 0 2

0

cosf

3 2

cosq

7 6 0 sinf 5 4 sinq 1

sinf cosq ¼ 4 cosf cosq sinq

0 sinf sinq cosf sinq cosq

0

sinq cosq 0 3

0

3

7 05 1

(B.7)

cosf sinf 5 0

Displacement transformation uR ¼ u sinf cosq þ v sinf sinq þ w cosf uf ¼ u cosf cosq þ v cosf sinq  w sinf uq ¼ u sinq þ v cosq

(B.8)

540

APPENDIX B Transformation of Field Variables

Stress transformation sR ¼ sx sin2 f cos2 q þ sy sin2 f sin2 q þ sz cos2 f þ2sxy sin2 f sinq cosq þ 2syz sinf cosf sinq þ 2szx sinf cosf cosq sf ¼ sx cos2 f cos2 q þ sy cos2 f sin2 q þ sz sin2 f þ 2sxy cos2 f sinq cosq  2syz sinf cosf sinq  2szx sinf cosf cosq sq ¼ sx sin2 q þ sy cos2 q  2sxy sinq cosq sRf ¼ sx sinf cosf cos2 q þ sy sinf cosf sin2 q  sz sinf cosf   þ 2sxy sinf cosf sinq cosq  syz sin2 f  cos2 f sinq    szx sin2 f  cos2 f cosq   sfq ¼ sx cosf sinq cosq þ sy cosf sinq cosq þ sxy cosf cos2 q  sin2 q

(B.9)

 syz sinf cosq þ szx sinf sinq

  sqR ¼ sx sinf sinq cosq þ sy sinf sinq cosq þ sxy sinf cos2 q  sin2 q þ syz cosf cosq  szx cosf sinq Inverse transformations of these results can be computed by formally inverting the system equations or redeveloping the results using tensor transformation theory.