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The influence of an initial twisting on tapered beams undergoing large displacements

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Abstract

The behaviour of pre-twisted and tapered beams (such as turbine or helicopter blades) is characterized by stress distributions that may be quite different from those of the usual beam theory, yielding couplings among bending, twisting and traction. We propose a physical–mathematical model for tapered beams that accounts for the effects of the pre-twist of the cross-sections along the centre-line. The beam centre-line may undergo large displacements, while its cross-sections see small warping both in- and out of their plane. Supposing infinitesimal strain, a variational approach provides the field equations, which are perturbed in terms of a small geometric ratio and shall be solved numerically in general. However, analytical closed-form solutions exist in some cases, such as for isotropic beams with pre-twisted, bi-tapered elliptic cross-sections; they are presented and compared with the results of nonlinear 3D-FEM simulations.

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Funding

The financial support of University "La Sapienza" of Roma (Grant No. RM11916B7ECCFCBF) and PRIN MIUR "Integrated mechanobiology approaches for a precise medicine in cancer treatment" is gratefully acknowledged.

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Appendix

Appendix

Equation (19) provide the components Fi, Mi of the stress resultants with respect to the current local triad ai, obtained by combining Eqs. (15)–(18), which provide the strain fields (15), with Eqs. (8)–(9), which relate the stress resultants (9) and the strain fields (15):

$$\begin{aligned} F_{1} & = Y\int_{\Sigma } {k_{2} x_{3} - k_{3} x_{2} + \gamma_{1} } + Y\int_{\Sigma } {e_{1,1} } + Y\int_{\Sigma } {k_{B1} \left( {x_{3} e_{1,2} - x_{2} e_{1,3} } \right)} \\ F_{2} & = G\int_{\Sigma } {e_{1,2} - k_{1} x_{3} } + G\,\int_{\Sigma } {2(1 + \nu )\left( {k_{2} x_{3} - k_{3} x_{2} + \gamma_{1} } \right)\left( {\Lambda_{2}^{ - 1} \Lambda^{\prime}_{2} x_{2} - k_{B1} x_{3} } \right)} + G\int_{\Sigma } {e_{2} } \\ F_{3} & = G\int_{\Sigma } {e_{1,3} + k_{1} x_{2} } + G\,\int_{\Sigma } {2(1 + \nu )\left( {k_{2} x_{3} - k_{3} x_{2} + \gamma_{1} } \right)\left( {\Lambda_{3}^{ - 1} \Lambda^{\prime}_{3} x_{3} + k_{B1} x_{2} } \right)} + G\int_{\Sigma } {e_{3} } \\ M_{1} & = G\int_{\Sigma } {x_{2} \left( {e_{1,3} + k_{1} x_{2} } \right) - x_{3} \left( {e_{1,2} - k_{1} x_{3} } \right)} + G\int_{\Sigma } {x_{2} e_{3} - x_{3} e_{2} } + \cdots \\ & \quad + \,G\,\int_{\Sigma } {2(1 + \nu )\left( {k_{2} x_{3} - k_{3} x_{2} + \gamma_{1} } \right)\left[ {x_{2} \left( {\Lambda_{3}^{ - 1} \Lambda^{\prime}_{3} x_{3} + k_{B1} x_{2} } \right) - x_{3} \left( {\Lambda_{2}^{ - 1} \Lambda^{\prime}_{2} x_{2} - k_{B1} x_{3} } \right)} \right]} \\ M_{2} & = Y\int_{\Sigma } {k_{2} x_{3}^{2} - k_{3} x_{3} x_{2} + \gamma_{1} x_{3} } + Y\,\int_{\Sigma } {x_{3} e_{1,1} } + Y\,\int_{\Sigma } {k_{B1} x_{3} (x_{3} e_{1,2} - x_{2} e_{1,3} )} \\ M_{3} & = Y\,\int_{\Sigma } {k_{3} x_{2}^{2} - k_{2} x_{2} x_{3} - \gamma_{1} x_{2} } - Y\,\int_{\Sigma } {x_{2} e_{1,1} } - Y\,\int_{\Sigma } {k_{B1} x_{2} (x_{3} e_{1,2} - x_{2} e_{1,3} )} \\ \end{aligned}$$
(23)

Performing the integrals with respect to the cross-section centroid, and considering that e1, e2, e3 satisfy Eqs. (16)–(18), we get Eq. (19), where the coefficients multiplying 1D strains and their s-derivatives are defined as

$$A = \int_{\Sigma } 1$$
(24)
$$J_{0} = \int_{\Sigma } {x_{2}^{2} + x_{3}^{2} }$$
(25)
$$J_{1} = \int_{\Sigma } {(e_{1,3}^{{k_{1} }} + x_{2} )^{2} + (e_{1,2}^{{k_{1} }} - x_{3} )^{2} }$$
(26)
$$J_{2} = \int_{\Sigma } {x_{3}^{2} }$$
(27)
$$J_{3} = \int_{\Sigma } {x_{2}^{2} }$$
(28)
$$J_{23} = \int_{\Sigma } {x_{2} x_{3} }$$
(29)
$$I_{2} = J^{\prime}_{2} + k_{B1} J_{23}$$
(30)
$$I_{3} = J^{\prime}_{3} - k_{B1} J_{23}$$
(31)
$$I_{23} = J^{\prime}_{23} - k_{B1} J_{2}$$
(32)
$$I_{32} = J^{\prime}_{23} + k_{B1} J_{3}$$
(33)
$$X_{1} = \int_{\Sigma } {e_{1}^{{k_{1} }} }$$
(34)
$$Z_{1} = \int_{\Sigma } {e_{1,1}^{{k_{1} }} } + k_{B1} (J_{0} - J_{1} )$$
(35)
$$X_{2} = \int_{\Sigma } {x_{3} e_{1}^{{k_{1} }} }$$
(36)
$$Z_{2} = \int_{\Sigma } {x_{3} e_{1,1}^{{k_{1} }} } + k_{B1} \int_{\Sigma } {x_{3} (x_{3} e_{1,2}^{{k_{1} }} - x_{2} e_{1,3}^{{k_{1} }} )}$$
(37)
$$X_{3} = \int_{\Sigma } {x_{2} e_{1}^{{k_{1} }} }$$
(38)
$$Z_{3} = \int_{\Sigma } {x_{2} e_{1,1}^{{k_{1} }} } + k_{B1} \int_{\Sigma } {x_{2} (x_{3} e_{1,2}^{{k_{1} }} - x_{2} e_{1,3}^{{k_{1} }} )}$$
(39)
$$V_{1} = \rho^{\prime}\rho^{ - 1} \int_{\Sigma } {x_{2} x_{3} } + k_{B1} (J_{0} - J_{1} )$$
(40)
$$V_{2} = \rho^{\prime}\rho^{ - 1} \int_{\Sigma } {x_{2} x_{3}^{2} } + k_{B1} \int_{\Sigma } {x_{3} (x_{2}^{2} + x_{3}^{2} )} + \int_{\Sigma } {x_{2} e_{3}^{{k_{2} }} - x_{3} e_{2}^{{k_{2} }} }$$
(41)
$$V_{3} = \rho^{\prime}\rho^{ - 1} \int_{\Sigma } {x_{3} x_{2}^{2} } + k_{B1} \int_{\Sigma } {x_{2} (x_{2}^{2} + x_{3}^{2} )} + \int_{\Sigma } {x_{3} e_{2}^{{k_{3} }} - x_{2} e_{3}^{{k_{3} }} }$$
(42)
$${\rm H}_{2} = \,\int_{\Sigma } {x_{2} e_{3}^{{k^{\prime}_{2} }} - x_{3} e_{2}^{{k^{\prime}_{2} }} }$$
(43)
$${\rm H}_{3} = \,\int_{\Sigma } {x_{3} e_{2}^{{k^{\prime}_{3} }} - x_{2} e_{3}^{{k^{\prime}_{3} }} }$$
(44)

where \(e_{j}^{{k_{i} }}\) and \(e_{j}^{{k^{\prime}_{i} }}\) are the values of \(e_{j}\) obtained by solving Eqs. (16)–(18) in the cases in which the quantity in superscript (e.g. \(k_{2}\)) is unitary and the others (e.g. \(k_{3}\), \(k^{\prime}_{2}\), \(k^{\prime}_{3}\), \(\gamma_{1}\)) are zero.

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Migliaccio, G., Ruta, G. The influence of an initial twisting on tapered beams undergoing large displacements. Meccanica 56, 1831–1845 (2021). https://doi.org/10.1007/s11012-021-01334-2

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