Appendices
1. Block structure of the matrix of form P whose elements appear in Eqs. (14):
$$ \mathbf{P}=\left[{\mathbf{P}}_{\mathrm{I}}\kern0.5em {\mathbf{P}}_{\mathrm{I}\mathrm{I}}\kern0.5em {\mathbf{P}}_{\mathrm{I}\mathrm{I}\mathrm{I}}\kern0.5em {\mathbf{P}}_{\mathrm{I}\mathrm{V}}\right]. $$
Here, P
I, P
II, P
III, and P
IV are 5 ×5 submatrices with the following structure and components:
$$ \begin{array}{c}\hfill {\mathbf{P}}_{\mathrm{I}}=\left[\begin{array}{ccccc}\hfill {p}_{11}\hfill & \hfill {p}_{12}\hfill & \hfill {p}_{13}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {p}_{21}\hfill & \hfill {p}_{22}\hfill & \hfill {p}_{21}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {p}_{31}\hfill & \hfill {p}_{32}\hfill & \hfill {p}_{33}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {p}_{44}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {p}_{55}\hfill \end{array}\right],\kern0.5em {\mathbf{P}}_{\mathrm{I}\mathrm{I}}=\left[\begin{array}{ccccc}\hfill {p}_{16}\hfill & \hfill {p}_{17}\hfill & \hfill {p}_{18}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {p}_{26}\hfill & \hfill {p}_{27}\hfill & \hfill {p}_{28}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {p}_{36}\hfill & \hfill {p}_{37}\hfill & \hfill {p}_{38}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {p}_{49}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {p}_{5,10}\hfill \end{array}\right],\hfill \\ {}\hfill {\mathbf{P}}_{\mathrm{I}\mathrm{I}\mathrm{I}}=\left[\begin{array}{ccccc}\hfill {p}_{1,11}\hfill & \hfill {p}_{1,12}\hfill & \hfill {p}_{1,13}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {p}_{2,11}\hfill & \hfill {p}_{2,12}\hfill & \hfill {p}_{2,13}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {p}_{3,11}\hfill & \hfill {p}_{3,12}\hfill & \hfill {p}_{3,13}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {p}_{4,14}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {p}_{5,15}\hfill \end{array}\right],\kern0.5em {\mathbf{P}}_{\mathrm{I}\mathrm{V}}=\left[\begin{array}{ccccc}\hfill {p}_{1,16}\hfill & \hfill {p}_{1,17}\hfill & \hfill {p}_{1,18}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {p}_{2,16}\hfill & \hfill {p}_{2,17}\hfill & \hfill {p}_{2,18}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {p}_{3,16}\hfill & \hfill {p}_{3,17}\hfill & \hfill {p}_{3,18}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {p}_{4,19}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {p}_{5,20}\hfill \end{array}\right],\hfill \\ {}\hfill {p}_{11}=1-\frac{xy}{ab}+\frac{3{x}^2y}{a^2b}+\frac{3x{y}^2}{a{b}^2}-\frac{2{x}^3y}{a^3b}-\frac{2x{y}^3}{b^3a}-\frac{3{x}^2}{a^2}-\frac{3{y}^2}{b^2}+\frac{2{x}^3}{a^3}+\frac{2{y}^3}{b^3}.\hfill \\ {}\hfill \begin{array}{cc}\hfill {p}_{12}=\frac{2{x}^2y}{ab}-\frac{x^3y}{a^3b}-\frac{xy}{b}+\frac{x^3}{a^2}-\frac{2{x}^2}{a}+x,\hfill & \hfill {p}_{13}=\frac{2x{y}^2}{ab}-\frac{x{y}^3}{a{b}^2}-\frac{2{y}^2}{b}+\frac{xy}{a}-\frac{y^3}{b^2}+y,\hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {p}_{16}=\frac{xy}{ab}+\frac{3{x}^2y}{a^2b}-\frac{3x{y}^2}{a{b}^2}+\frac{2{x}^3y}{a^3b}+\frac{2x{y}^3}{b^3a}+\frac{3{x}^2}{a^2}-\frac{2{x}^3}{a^3},\hfill & \hfill {p}_{17}=\frac{x^2y}{ab}-\frac{x^3y}{a^2b}+\frac{x^3}{a^2}-\frac{x^2}{a},\hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill {p}_{18}=-\frac{2x{y}^2}{ab}+\frac{x{y}^3}{a{b}^2}+\frac{xy}{a},\hfill & \hfill {p}_{1,11}=-\frac{xy}{ab}+\frac{3{x}^2y}{a^2b}+\frac{3x{y}^2}{a{b}^2}-\frac{2{x}^3y}{a^3b}-\frac{2x{y}^3}{b^3a},\hfill & \hfill {p}_{1,12}=-\frac{x^2y}{ab}+\frac{x^3y}{a^2b},\hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {p}_{1,13}=-\frac{x{y}^2}{ab}+\frac{x{y}^3}{a{b}^2},\hfill & \hfill {p}_{1,16}=\frac{xy}{ab}-\frac{3{x}^2y}{a^2b}-\frac{3x{y}^2}{a{b}^2}+\frac{2{x}^3y}{a^3b}+\frac{2x{y}^3}{b^3a}+\frac{3{y}^2}{b^2}-\frac{2{y}^3}{b^3},\hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {p}_{1,17}=-\frac{2{x}^2y}{ab}+\frac{x^3y}{a^2b}+\frac{xy}{b},\hfill & \hfill {p}_{1,18}=\frac{x{y}^2}{ab}-\frac{x{y}^3}{a{b}^2}-\frac{y^2}{b}+\frac{y^3}{b^2},\hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {p}_{21}=-\frac{6{x}^2y}{a^3b}+\frac{6xy}{a^2b}-\frac{y}{ab}+\frac{3{y}^2}{a{b}^2}-\frac{2{y}^3}{b^3a}-\frac{6x}{a^2}+\frac{6{x}^2}{a^3},\hfill & \hfill {p}_{22}=1+\frac{4xy}{ab}-\frac{3{x}^2y}{a^2b}\hfill \end{array}+\frac{3{x}^2}{a^2}-\frac{4x}{a}-\frac{y}{b},\hfill \\ {}\hfill \begin{array}{ccc}\hfill {p}_{23}=\frac{2{y}^2}{ab}-\frac{y^3}{a{b}^2}-\frac{y}{a},\hfill & \hfill {p}_{26}=\frac{6{x}^2y}{a^3b}-\frac{6xy}{a^2b}+\frac{y}{ab}-\frac{3{y}^2}{a{b}^2}+\frac{2{y}^3}{b^3a}+\frac{6x}{a^2}-\frac{6{x}^2}{a^3},\hfill & \hfill {p}_{27}=\frac{2xy}{ab}-\frac{3{x}^2y}{a^2b}+\frac{3{x}^2}{a^2}-\frac{2x}{a}\hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill {p}_{28}=-\frac{2{y}^2}{ab}+\frac{y^3}{a{b}^2}+\frac{y}{a},\hfill & \hfill {p}_{2,11}=-\frac{6{x}^2y}{a^3b}+\frac{6xy}{a^2b}-\frac{y}{ab}+\frac{3{y}^2}{a{b}^2}-\frac{2{y}^3}{b^3a},\hfill & \hfill {p}_{2,12}=-\frac{2xy}{ab}+\frac{3{x}^2y}{a^2b},\hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {p}_{2,13}=-\frac{y^2}{ab}+\frac{y^3}{a{b}^2},\hfill & \hfill {p}_{2,16}=-\frac{6{x}^2y}{a^3b}-\frac{6xy}{a^2b}+\frac{y}{ab}-\frac{3{y}^2}{a{b}^2}+\frac{2{y}^3}{b^3a},\hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill {p}_{2,17}=-\frac{4xy}{ab}+\frac{3{x}^2y}{a^2b}+\frac{y}{b},\hfill & \hfill {p}_{2,18}=\frac{y^2}{ab}-\frac{y^3}{a{b}^2},\hfill & \hfill {p}_{31}=-\frac{6x{y}^2}{b^3a}-\frac{x}{ab}+\frac{3{x}^2}{a^2b}-\frac{2{x}^3}{a^3b}-\frac{6y}{b^2}+\frac{6{y}^2}{b^3}+\frac{6xy}{a{b}^2},\hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill {p}_{32}=\frac{2{x}^2}{ab}-\frac{x^3}{a^2b}-\frac{x}{b},\hfill & \hfill {p}_{33}=1+\frac{4xy}{ab}-\frac{3x{y}^2}{a{b}^2}+\frac{3{y}^2}{b^2}-\frac{x}{a}-\frac{4y}{b},\hfill & \hfill {p}_{36}=\frac{6x{y}^2}{b^3a}+\frac{x}{ab}-\frac{3{x}^2}{a^2b}+\frac{2{x}^3}{a^3b}-\frac{6xy}{a{b}^2},\hfill \end{array}\hfill \\ {}\hfill \begin{array}{cccc}\hfill {p}_{37}=\frac{x^2}{ab}-\frac{x^3}{a^2b},\hfill & \hfill {p}_{38}=\frac{4xy}{ab}+\frac{3x{y}^2}{a{b}^2}+\frac{x}{a},\hfill & \hfill {p}_{3,11}=-\frac{6x{y}^2}{b^3a}-\frac{x}{ab}+\frac{3{x}^2}{a^2b}-\frac{2{x}^3}{a^3b}+\frac{6xy}{a{b}^2},\hfill & \hfill {p}_{3,12}=-\frac{x^2}{ab}+\frac{x^3}{a^2b},\hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccc}\hfill {p}_{3,13}=-\frac{2xy}{ab}+\frac{3x{y}^2}{a{b}^2},\hfill & \hfill {p}_{3,16}=\frac{6x{y}^2}{b^2a}+\frac{x}{ab}-\frac{3{x}^2}{a^2b}+\frac{2{x}^3}{a^3b}+\frac{6y}{b^2}-\frac{6{y}^2}{b^3}-\frac{6xy}{a{b}^2},\hfill & \hfill {p}_{3,17}=-\frac{2{x}^2}{ab}+\frac{x^3}{a^2b}+\frac{x}{b},\hfill \end{array}\hfill \\ {}\hfill \begin{array}{cccc}\hfill {p}_{3,18}=\frac{2xy}{ab}-\frac{3x{y}^2}{a{b}^2}+\frac{3{y}^2}{b^2}-\frac{2y}{b},\hfill & \hfill {p}_{44}=1+\frac{xy}{ab}-\frac{x}{a}-\frac{y}{b},\hfill & \hfill {p}_{49}=-\frac{xy}{ab}+\frac{x}{a},\hfill & \hfill {p}_{4,14}=\frac{xy}{ab},\hfill \end{array}\hfill \\ {}\hfill \begin{array}{ccccc}\hfill {p}_{4,19}=-\frac{xy}{ab}+\frac{y}{b},\hfill & \hfill {p}_{55}=1+\frac{xy}{ab}-\frac{x}{a}-\frac{y}{b},\hfill & \hfill {p}_{5,10}=-\frac{xy}{ab}+\frac{x}{a},\hfill & \hfill {p}_{5,15}=\frac{xy}{ab},\hfill & \hfill {p}_{5,20}=-\frac{xy}{ab}+\frac{y}{b}.\hfill \end{array}\hfill \end{array} $$
2. Components of the matrix of geometrical rigidity of the element R
e
:
$$ \begin{array}{c}\hfill {r}_{11}=\frac{46a}{105b},\kern0.5em {r}_{12}=\frac{b}{30},\kern0.5em {r}_{13}=\frac{11{b}^2}{210a},\kern0.5em {r}_{16}=-\frac{46b}{105a},\kern0.5em {r}_{17}=\frac{b}{30},\kern0.5em {r}_{18}=\frac{11{b}^2}{210a},\kern0.5em {r}_{1,11}=-\frac{17b}{105a},\hfill \\ {}\hfill {r}_{1,12}=\frac{b}{60},\kern0.5em {r}_{1,13}=\frac{13{b}^2}{420a},\kern0.5em {r}_{1,16}=\frac{17b}{105a},\kern0.5em {r}_{1,17}=\frac{b}{60},\kern0.5em {r}_{1,18}=-\frac{13{b}^2}{420a},\kern0.5em {r}_{22}=\frac{2 ab}{45},\kern0.5em {r}_{26}=-\frac{b}{30},\kern0.5em {r}_{27}=-\frac{ab}{90},\kern0.5em {r}_{2,11}=-\frac{b}{60},\hfill \\ {}\hfill {r}_{2,12}=-\frac{ab}{180},\kern0.5em {r}_{2,16}=\frac{b}{60},\kern0.5em {r}_{2,17}=\frac{ab}{45},\kern0.5em {r}_{33}=\frac{b^3}{105a},\kern0.5em {r}_{36}=-\frac{11{b}^2}{210a},\kern0.5em {r}_{38}=-\frac{b^3}{105a},\kern0.5em {r}_{3,11}=-\frac{13{b}^2}{420a},\kern0.5em {r}_{3,13}=\frac{b^3}{140a},\hfill \\ {}\hfill {r}_{3,16}=\frac{13{b}^2}{420a},\kern0.5em {r}_{3,18}=\frac{b^3}{140a},\kern0.5em {r}_{66}=\frac{46b}{405a},\kern0.5em {r}_{67}=-\frac{b}{30},\kern0.5em {r}_{68}=\frac{11{b}^2}{210a},\kern0.5em {r}_{6,11}=\frac{17b}{105a},\kern0.5em {r}_{6,12}=-\frac{b}{60},\kern0.5em {r}_{6,13}=-\frac{13{b}^2}{420a},\hfill \\ {}\hfill {r}_{6,16}=-\frac{17b}{105a},\kern0.5em {r}_{6,17}=-\frac{b}{60},\kern0.5em {r}_{6,18}=\frac{13{b}^2}{420a},\kern0.5em {r}_{77}=\frac{2 ab}{45},\kern0.5em {r}_{7,11}=-\frac{b}{60},\kern0.5em {r}_{7,12}=\frac{ab}{45},\kern0.5em {r}_{7,16}=\frac{b}{60},\kern0.5em {r}_{7,17}=-\frac{ab}{180},\hfill \\ {}\hfill {r}_{88}=\frac{b^3}{105a},\kern0.5em {r}_{8,11}=\frac{13{b}^2}{420a},\kern0.5em {r}_{8,13}=-\frac{b^3}{140a},\kern0.5em {r}_{8,16}=-\frac{13{b}^2}{420a},\kern0.5em {r}_{8,18}=\frac{b^3}{140a},\kern0.5em {r}_{11,11}=\frac{46b}{105a},\kern0.5em {r}_{11,12}=-\frac{b}{30},\hfill \\ {}\hfill {r}_{11,13}=-\frac{11{b}^2}{210a},\kern0.5em {r}_{11,16}=-\frac{46b}{105a},\kern0.5em {r}_{11,17}=-\frac{b}{30},\kern0.5em {r}_{11,18}=\frac{11{b}^2}{201a},\kern0.5em {r}_{12,12}=\frac{2 ab}{45},\kern0.5em {r}_{12,16}=\frac{b}{30},\kern0.5em {r}_{12,17}=-\frac{ab}{90},\hfill \\ {}\hfill {r}_{13,13}=\frac{b^3}{105a},\kern0.5em {r}_{13,16}=\frac{11{b}^2}{201a},\kern0.5em {r}_{13,18}=-\frac{b^3}{105a},\kern0.5em {r}_{16,16}=\frac{46b}{105a},\kern0.5em {r}_{16,17}=\frac{b}{30},\kern0.5em {r}_{16,18}=-\frac{11{b}^2}{210a},\kern0.5em {r}_{17,17}=\frac{2 ab}{45},\kern0.5em {r}_{18,18}=\frac{b^3}{105a}.\hfill \end{array} $$
