A novel analytical solution for warping analysis of arbitrary annular wedge-shaped bars

Abstract

Structural components with arbitrary cross sections play a key role in several engineering fields. Many engineering structures are subjected to torsional moments, so it is important to design and analyze these type of crucial engineering issues. First, this study offers a novel analytical solution for Prandtl’s stress distribution of arbitrary annular wedge-shaped bars under uniform torsion moment based on eigenfunction expansion. Next, warping function is derived by integration of Cauchy-Riemann type relations in polar coordinate system. The solution encompasses existing solutions for standard wedge-shaped bars as subsets. Finally, accuracy of the proposed analytical method is fully demonstrated through some benchmarks which are available in the literature.

Appendices

Appendix A: Derivation of Eq. (2)

The method of eigenfunction expansion constructs a series solution for the stress function containing products of radial and angular components as given by:

$$\psi \left( {r,\theta } \right) = \mathop \sum \limits_{n = 1}^{\infty } U_{n} \left( r \right)\sin \left( {\omega \theta } \right)$$

(A.1)

Obviously the angular component obeys the boundary conditions along the straight edges $\theta = 0$ and $\theta = \beta$. Similarly, the RHS of Eq. (1) is expanded into a Fourier trigonometric series:

$$- 2 = - \frac{8}{\pi }\mathop \sum \limits_{n = 1,3,5 \ldots }^{\infty } \frac{{\sin \left( {\omega \theta } \right)}}{n}$$

(A.2)

Given the uniqueness of Fourier series coefficients and inserting (A.1) and (A.2) into the BVP formulated by Eq. (1), one arrives at a nonhomogeneous ODE for the radial component which applies to odd values of integer $n$:

$$\frac{{{\text{d}}^{2} U_{n} \left( r \right)}}{{{\text{d}}r^{2} }} + \frac{1}{r}\frac{{{\text{d}}U_{n} \left( r \right)}}{{{\text{d}}r}} - \left( {\frac{\omega }{r}} \right)^{2} U_{n} \left( r \right) = - \frac{8}{n\pi }$$

(A.3)

The method of variation of parameters presents the solution of (A.3) as follows:

$$U_{n} \left( r \right) = \frac{8}{{n\pi \left( {\omega^{2} - 4} \right)}}\left\{ {\frac{{\left[ {\left( \frac{a}{r} \right)^{\omega } - \left( \frac{r}{a} \right)^{\omega } } \right]b^{2} + \left[ {\left( \frac{r}{b} \right)^{\omega } - \left( \frac{b}{r} \right)^{\omega } } \right]a^{2} }}{{\left( \frac{b}{a} \right)^{\omega } - \left( \frac{a}{b} \right)^{\omega } }} + r^{2} } \right\}$$

(A.4)

which satisfies the boundary conditions along the curved edges $r = a$ and $r = b$. This completes the desired expression for the stress function in (A.1), leading to Eq. (2).

Appendix B: Derivation of Eq. (6)

The warping function and Prandtl’s stress function are interconnected through Cauchy-Riemann type equation, satisfying the following set of two linear PDEs in the Cartesian coordinates [21]:

$$\begin{gathered} \varphi_{,x} - y = \psi_{,y} \hfill \\ \varphi_{,y} + x = - \psi_{,x} \hfill \\ \end{gathered}$$

(B.1)

This is converted to the polar coordinates:

$$\begin{gathered} \varphi_{,r} = r^{ - 1} \psi_{,\theta } \hfill \\ \varphi_{,\theta } = - r\left( {\psi_{,r} + r} \right) \hfill \\ \end{gathered}$$

(B.2)

with the RHSs computed from Eqs. (4)-(5). To solve for the warping function, the first equation in (B.2) is integrated with respect to radial coordinate resulting in:

$$\varphi \left( {r,\theta } \right) = \frac{8}{\pi }\mathop \sum \limits_{n = 1,3,5 \ldots }^{\infty } \frac{{\cos \left( {\omega \theta } \right)}}{{n\left( {\omega^{2} - 4} \right)}}\left\{ {\frac{{\left[ {\left( \frac{r}{b} \right)^{\omega } + \left( \frac{b}{r} \right)^{\omega } } \right]a^{2} - \left[ {\left( \frac{r}{a} \right)^{\omega } + \left( \frac{a}{r} \right)^{\omega } } \right]b^{2} }}{{\left( \frac{b}{a} \right)^{\omega } - \left( \frac{a}{b} \right)^{\omega } }} + \frac{{\omega r^{2} }}{2}} \right\} + f\left( \theta \right)$$

(B.3)

where $f\left( \theta \right)$ arises as constant of integration because it is independent from $r$. Substituting (B.3) in the second equation in (B.2) after some algebra gives $f^{\prime}\left( \theta \right) = 0$ implying that $f\left( \theta \right)$ is arbitrary constant which is set equal to zero in the present calculations. Equation (6) expresses Eq. (B.3) in a rather compact form.