The cylindrical coordinate system
\(Or\varphi z\) has been used to formulate the governing equations of the uniform torsion problem of compound annular wedge-shape bar. The cross section
A can be divided into three parts as
\(A=A_{1}\cup A_{2}\cup A_{3}\), where
$$\begin{aligned}&A_{1}=\{ (r,\varphi ) | R_{0}\le r \le R_{1}, \, 0 \le \varphi \le \alpha \} \end{aligned}$$
(1)
$$\begin{aligned}&A_{2}=\{ (r,\varphi ) | R_{1}\le r \le R_{2}, \, 0 \le \varphi \le \alpha \} \end{aligned}$$
(2)
$$\begin{aligned}&A_{3}=\{ (r,\varphi ) | R_{2}\le r \le R_{3}, \, 0 \le \varphi \le \alpha \} \end{aligned}$$
(3)
There are perfect connections between the beam components whose cross sections are
\(A_{1}\),
\(A_{2}\) and
\(A_{3}\). From this fact it follows that axial displacement and radial shearing stress field are continuous on the whole cross section
A. The length of the compound bar is denoted by
L. The material of the beam component
\(B_{i}=A_{i}\times (0,L)\) \((i=1,2,3)\) is cylindrically orthotropic with shear modulus
\(G_{ir}\),
\(G_{i\varphi }\) \((i=1,2,3)\). In the present problem the Prandtl’s stress function formulation of the considered Saint-Venant torsion leads to the next coupled boundary-value problem [
2,
4‐
6,
9,
11,
12]
$$\begin{aligned}&\frac{\partial ^{2} U_{1}}{\partial r^{2}} + \frac{1}{r}\frac{\partial U_{1}}{\partial r}+\frac{g_{1}^{2}}{r^{2}}\frac{\partial ^{2} U_{1}}{\partial \varphi ^{2}} = -2 G_{1\varphi } \qquad g_{1}=\sqrt{\frac{G_{1\varphi }}{G_{1r}}} \quad \mathrm {in}\, A_{1} \end{aligned}$$
(4)
$$\begin{aligned}&U_{1}=0\quad R_{0} \le r \le R_{1} \quad \varphi =0 \quad \mathrm {and} \quad \varphi =\alpha \end{aligned}$$
(5)
$$\begin{aligned}&\frac{\partial ^{2} U_{2}}{\partial r^{2}} + \frac{1}{r}\frac{\partial U_{2}}{\partial r} + \frac{g_{2}^{2}}{r^{2}}\frac{\partial ^{2} U_{2}}{\partial \varphi ^{2}} = -2 G_{2\varphi } \qquad g_{2}=\sqrt{\frac{G_{2\varphi }}{G_{2r}}} \quad \mathrm {in}\,A_{2} \end{aligned}$$
(6)
$$\begin{aligned}&U_{2}=0\quad R_{1} \le r \le R_{2} \quad \varphi =0 \quad \mathrm {and} \quad \varphi =\alpha \end{aligned}$$
(7)
$$\begin{aligned}&\frac{\partial ^{2} U_{3}}{\partial r^{2}} + \frac{1}{r}\frac{\partial U_{3}}{\partial r} + \frac{g_{3}^{2}}{r^{2}}\frac{\partial ^{2} U_{3}}{\partial \varphi ^{2}} = -2 G_{3\varphi } \qquad g_{3}=\sqrt{\frac{G_{3\varphi }}{G_{3r}}} \quad \mathrm {in}\,A_{3} \end{aligned}$$
(8)
$$\begin{aligned}&U_{3}=0\quad R_{2} \le r \le R_{3} \quad \varphi =0 \quad \mathrm {and} \quad \varphi =\alpha \end{aligned}$$
(9)
$$\begin{aligned}&U_{1}(R_{0},\varphi )=0\qquad 0 \le \varphi \le \alpha \end{aligned}$$
(10)
$$\begin{aligned}&U_{3}(R_{3},\varphi )=0\qquad 0 \le \varphi \le \alpha \end{aligned}$$
(11)
$$\begin{aligned}&U_{1}(R_{1},\varphi ) = U_{2}(R_{1},\varphi ) \qquad 0\le \varphi \le \alpha \end{aligned}$$
(12)
$$\begin{aligned}&U_{2}(R_{2},\varphi ) = U_{3}(R_{2},\varphi ) \qquad 0\le \varphi \le \alpha \end{aligned}$$
(13)
$$\begin{aligned}&\frac{1}{G_{1\varphi }} \frac{\partial U_{1}}{\partial r} = \frac{1}{G_{2\varphi }} \frac{\partial U_{2}}{\partial r}\qquad r=R_{1} \qquad 0 \le \varphi \le \alpha \end{aligned}$$
(14)
$$\begin{aligned}&\frac{1}{G_{2\varphi }} \frac{\partial U_{2}}{\partial r} = \frac{1}{G_{3\varphi }} \frac{\partial U_{3}}{\partial r}\qquad r=R_{2} \qquad 0 \le \varphi \le \alpha \end{aligned}$$
(15)
Equations (
4), (
6) and (
8) formulate the strain compatibility conditions in terms of stress function
\(U_{i}=U_{i}(r,\varphi )\) \((i=1,2,3)\). The boundary conditions (
5), (
7), (
9), (
10) and (
11) express that the whole boundary contour of cross section
A is traction free. The continuity conditions of radial shearing stresses on the common boundary curve of
\(A_{1}\) and
\(A_{2}\) on the common boundary curve of
\(A_{2}\) and
\(A_{3}\) are formulated by Eqs. (
12) and (
13). Equations (
14) and (
15) provide the continuity of the axial displacement over the whole cross section
A. The relation between the Prandtl’s stress functions
\(U_{i}=U_{i}(r,\varphi )\) and torsion function
\(\omega _{i}=\omega _{i}(r,\varphi )\) are described by the following systems of equations [
2,
5,
6,
9,
11,
12]
$$\begin{aligned} G_{ir}\frac{\partial \omega _{i}}{\partial r} = \frac{1}{r}\frac{\partial U_{i}}{\partial \varphi }, \quad G_{i\varphi } \frac{\partial \omega _{i}}{\partial \varphi } = - r \frac{\partial U_{i}}{\partial r} - G_{i\varphi } r^{2} \qquad (r,\varphi )\in A_{i} \quad (i=1,2,3) \end{aligned}$$
(16)
Equation (
16) is based on formulae of shearing stresses
\(\tau _{irz}=\tau _{irz}(r,\varphi )\) and
\(\tau _{i\varphi z}=\tau _{i\varphi z}(r,\varphi )\) expressed in terms of
\(U_{i}=U_{i}(r,\varphi )\) and
\(\omega _{i}=\omega _{i}(r,\varphi )\) \((i=1,2,3)\) which are as follows
$$\begin{aligned}&\frac{\tau _{irz}}{\vartheta } = G_{ir} \frac{\partial \omega _{i}}{\partial r} = \frac{1}{r}\frac{\partial U_{i}}{\partial \varphi } \qquad (i=1,2,3) \end{aligned}$$
(17)
$$\begin{aligned}&\frac{\tau _{i\varphi z}}{\vartheta } = G_{i\varphi } \left( \frac{1}{r}\frac{\partial \omega _{i}}{\partial \varphi } + r\right) = -\frac{\partial U_{i}}{\partial r} \qquad (i=1,2,3) \end{aligned}$$
(18)
In Eqs. (
14), (
15)
\(\vartheta \) denotes the rate of twist with respect to the axial coordinate
z [
5,
6,
8]. The relation between the applied torque
T and
\(\vartheta \) is as follows
$$\begin{aligned} T=\vartheta S \end{aligned}$$
(19)
where
S is the torsional rigidity of the compound cross section
A. According to Prandtl’s formulation of uniform torsion we have [
5,
6,
11,
12]
$$\begin{aligned} S=2\left( \int \limits _{A_{1}} U_{1}\,\mathrm{d} A + \int \limits _{A_{2}} U_{2}\,\mathrm{d} A + \int \limits _{A_{3}} U_{3} \, \mathrm{d} A\right) . \end{aligned}$$
(20)
Here, we note for isotropic beam component the shear modulus in radial and circumferential direction is the same, that is
$$\begin{aligned} G_{r}=G_{\varphi }=G \end{aligned}$$
(21)