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A smeared stiffener model for vibration analysis of anisogrid-stiffened composite conical shells using differential quadrature method

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Abstract

Anisogrid-stiffened composite conical shells have been widely applied in aerospace engineering as highly efficient structural elements. In this paper, the smeared stiffener approach is adopted to analyze the free vibration behavior of stiffened composite conical shells. The stiffening elements are modeled as beams that can bear bending moments and shear forces along with the axial forces. The stiffeners' stiffness is superimposed onto those of the shell, resulting in a variable equivalent stiffness for the entire structure. According to the Donnell-type shell model and Hamilton’s law, the kinetic and elastic energies are formulated, and then, the coupled partial differential equations of the anisogrid-stiffened composite conical shells are derived. Using the differential quadrature method (DQM) and appropriate harmonic functions, the equations are numerically solved in meridional and circumferential directions. Validation of the results is accomplished by making a comparison between the obtained results and those of other studies. Finally, several of the essential design variables are analyzed mathematically for various boundary conditions, and conclusions are drawn based on these findings. It is found that the influence of the boundary condition at the small end of the conical shell on the natural frequency of the anisogrid-stiffened structure is greater compared to the un-stiffened structure.

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Authors and Affiliations

  1. Flight & Engineering Department, Imam Ali University, Tehran, Iran

    Ali Asghar Naderi, Hussein Pachdaman & Alireza Zandieh

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  1. Ali Asghar Naderi
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Contributions

AAN and HP involved in simulation and figure preparation; AAN and HP involved in extraction of the formulations/relations; AAN contributed to conceptualization; AAN and AZ involved in writing and editing; AAN and AZ involved in supervision; and AAN involved in project administration. All authors have read and agreed to the final version of the manuscript.

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Correspondence to Ali Asghar Naderi.

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Appendix

Appendix

In Eq. (39), Elements \({\Gamma }_{11}-{\Gamma }_{55}\) are related with the conditions at \(x={x}_{0}\), which are defined as:

Clamped (C):

$$\begin{gathered} \Gamma_{11} = \Gamma_{22} = \Gamma_{33} = \Gamma_{44} = \Gamma_{55} = I_{1} , \hfill \\ \Gamma_{12} = \Gamma_{13} = \Gamma_{14} = \Gamma_{15} = \Gamma_{21} = \Gamma_{23} = \Gamma_{24} = \Gamma_{25} = \Gamma_{31} = \Gamma_{32} = \Gamma_{34} = \Gamma_{35} = \Gamma_{41} = \Gamma_{42} = \Gamma_{43} \hfill \\ = \Gamma_{45} = \Gamma_{51} = \Gamma_{52} = \Gamma_{53} = \Gamma_{54} = \left\{ 0 \right\}_{1 \times N} , \hfill \\ \end{gathered}$$
(45)

Simply supported (S):

$$\begin{gathered} \Gamma_{11} = A_{11,11} A_{1} + A_{12,11} \sin \alpha \,a_{1,1} ,\,\,\,\,\,\,\,\,\Gamma_{14} = \Gamma_{41} = B_{11,11} A_{1} + B_{12,11} \sin \alpha \,a_{1,1} ,\,\,\,\,\,\,\,\, \hfill \\ \Gamma_{44} = D_{11,11} A_{1} + D_{12,11} \sin \alpha \,a_{1,1} ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Gamma_{22} = \Gamma_{33} = \Gamma_{55} = I_{1} ,\,\,\,\,\,\,\,\,\,\,\Gamma_{12} = \Gamma_{13} = \Gamma_{15} = \Gamma_{21} = \Gamma_{23} \hfill \\ = \Gamma_{24} = \Gamma_{25} = \Gamma_{31} = \Gamma_{32} = \Gamma_{34} = \Gamma_{35} = \Gamma_{42} = \Gamma_{43} = \Gamma_{45} = \Gamma_{51} = \Gamma_{52} = \Gamma_{53} = \Gamma_{54} = \left\{ 0 \right\}_{1 \times N} , \hfill \\ \end{gathered}$$
(46)

Free (F):

$$\begin{aligned} \Gamma_{11} = & A_{11,11} A_{1} + A_{12,11} \sin \alpha \,a_{1,1} ,\,\,\,\,\,\,\,\,\Gamma_{12} = nA_{12,11} a_{1,1} ,\,\,\,\,\,\,\,\Gamma_{13} = A_{12,11} \cos \alpha \,a_{1,1} ,\,\,\,\,\,\,\,\Gamma_{14} = B_{11,11} A_{1} + B_{12,11} \sin \alpha \,a_{1,1} , \\ \Gamma_{15} = & nB_{12,11} a_{1,1} ,\,\,\,\,\,\,\,\Gamma_{21} = - nA_{66,11} a_{1,1} ,\,\,\,\,\,\,\,\,\Gamma_{22} = A_{66,11} A_{1} - A_{66,11} \sin \alpha \,a_{1,1} ,\,\,\,\,\,\,\Gamma_{23} = \left\{ 0 \right\}_{1 \times N} ,\,\,\,\,\,\,\Gamma_{24} = - nB_{66,11} a_{1,1} \\ \Gamma_{25} = & B_{66,11} A_{1} - B_{66,11} \sin \alpha \,a_{1,1} ,\,\,\,\,\,\,\,\Gamma_{31} = \Gamma_{32} = \Gamma_{35} = \left\{ 0 \right\}_{1 \times N} ,\,\,\,\,\,\,\,\,\Gamma_{33} = A_{1} ,\,\,\,\,\,\,\Gamma_{34} = I_{1} ,\,\,\,\,\,\,\,\,\Gamma_{41} = B_{11,11} A_{1} + B_{12,11} \sin \alpha \,a_{1,1} , \\ \Gamma_{42} = & nB_{12,11} a_{1,1} ,\,\,\,\,\,\,\,\Gamma_{43} = B_{12,11} \cos \alpha \,a_{1,1} ,\,\,\,\,\,\,\,\Gamma_{44} = D_{11,11} A_{1} + D_{12,11} \sin \alpha \,a_{1,1} ,\,\,\,\,\,\,\,\,\Gamma_{45} = nD_{12,11} \,a_{1,1} , \\ \Gamma_{51} = & - nB_{66,11} a_{1,1} ,\,\,\,\,\Gamma_{52} = B_{66,11} A_{1} - B_{66,11} \sin \alpha \,a_{1,1} ,\,\,\,\,\,\Gamma_{53} = \left\{ 0 \right\}_{1 \times N} ,\,\,\,\,\,\Gamma_{54} = - nD_{66,11} \,a_{1,1} ,\,\,\,\,\,\Gamma_{55} = D_{66,11} A_{1} - D_{66,11} \sin \alpha \,a_{1,1} , \\ \end{aligned}$$
(47)

And elements \({\Gamma }_{61}-{\Gamma }_{105}\) are related with the conditions at \(x={x}_{0}+L\), which are defined as:

Clamped (C):

$$\begin{gathered} \Gamma_{61} = \Gamma_{72} = \Gamma_{83} = \Gamma_{94} = \Gamma_{105} = I_{N} , \hfill \\ \Gamma_{62} = \Gamma_{63} = \Gamma_{64} = \Gamma_{65} = \Gamma_{71} = \Gamma_{73} = \Gamma_{74} = \Gamma_{75} = \Gamma_{81} = \Gamma_{82} = \Gamma_{84} = \Gamma_{85} = \Gamma_{91} = \Gamma_{92} = \Gamma_{93} \hfill \\ = \Gamma_{95} = \Gamma_{101} = \Gamma_{102} = \Gamma_{103} = \Gamma_{104} = \left\{ 0 \right\}_{1 \times N} , \hfill \\ \end{gathered}$$
(48)

Simply supported (S):

$$\begin{gathered} \Gamma_{61} = A_{11,NN} A_{N} + A_{12,N} \sin \alpha \,a_{1,N} ,\,\,\,\,\,\,\,\,\Gamma_{64} = \Gamma_{91} = B_{11,NN} A_{N} + B_{12,NN} \sin \alpha \,a_{1,N} ,\,\,\,\,\,\,\,\, \hfill \\ \Gamma_{94} = D_{11,NN} A_{N} + D_{12,NN} \sin \alpha \,a_{1,N} ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Gamma_{72} = \Gamma_{83} = \Gamma_{105} = I_{1} ,\,\,\,\,\,\,\,\,\,\Gamma_{62} = \Gamma_{63} = \Gamma_{65} = \Gamma_{71} = \Gamma_{73} \hfill \\ = \Gamma_{74} = \Gamma_{75} = \Gamma_{81} = \Gamma_{82} = \Gamma_{84} = \Gamma_{85} = \Gamma_{92} = \Gamma_{93} = \Gamma_{95} = \Gamma_{101} = \Gamma_{102} = \Gamma_{103} = \Gamma_{104} = \left\{ 0 \right\}_{1 \times N} , \hfill \\ \end{gathered}$$
(49)

Free (F):

$$\begin{gathered} \Gamma_{61} = A_{11,NN} A_{N} + A_{12,NN} \sin \alpha \,a_{1,N} ,\,\,\,\,\,\,\,\,\Gamma_{62} = nA_{12,NN} a_{1,N} ,\,\,\,\,\,\,\,\Gamma_{63} = A_{12,NN} \cos \alpha \,a_{1,N} ,\,\,\,\,\,\,\,\Gamma_{64} = B_{11,NN} A_{N} + B_{12,NN} \sin \alpha \,a_{1,N} , \hfill \\ \Gamma_{65} = nB_{12,NN} a_{1,N} ,\,\,\,\,\,\,\,\Gamma_{71} = - nA_{66,NN} a_{1,N} ,\,\,\,\,\,\,\,\,\Gamma_{72} = A_{66,NN} A_{N} - A_{66,NN} \sin \alpha \,a_{1,N} ,\,\,\,\,\,\,\Gamma_{73} = \left\{ 0 \right\}_{1 \times N} ,\,\,\,\,\,\,\Gamma_{74} = - nB_{66,NN} a_{1,N} \hfill \\ \Gamma_{75} = B_{66,NN} A_{N} - B_{66,NN} \sin \alpha \,a_{1,N} ,\,\,\,\,\,\Gamma_{81} = \Gamma_{82} = \Gamma_{85} = \left\{ 0 \right\}_{1 \times N} ,\,\,\,\,\,\Gamma_{83} = A_{1} ,\,\,\,\,\,\Gamma_{84} = I_{1} ,\,\,\,\,\,\Gamma_{91} = B_{11,NN} A_{N} + B_{12,NN} \sin \alpha \,a_{1,N} , \hfill \\ \Gamma_{92} = nB_{12,NN} a_{1,N} ,\,\,\,\,\,\Gamma_{93} = B_{12,NN} \cos \alpha \,a_{1,N} ,\,\,\,\,\Gamma_{94} = D_{11,11} A_{N} + D_{12,NN} \sin \alpha \,a_{1,N} ,\,\,\,\,\Gamma_{95} = nD_{12,NN} \,a_{1,N} ,\,\,\,\,\Gamma_{101} = - nB_{66,NN} a_{1,N} ,\, \hfill \\ \Gamma_{102} = B_{66,NN} A_{N} - B_{66,NN} \sin \alpha \,a_{1,N} ,\,\,\,\,\,\,\,\,\Gamma_{103} = \left\{ 0 \right\}_{1 \times N} ,\,\,\,\,\,\,\,\Gamma_{104} = - nD_{66,NN} \,a_{1,N} ,\,\,\,\,\,\,\Gamma_{105} = D_{66,NN} A_{N} - D_{66,NN} \sin \alpha \,a_{1,N} , \hfill \\ \end{gathered}$$
(50)

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Naderi, A.A., Pachdaman, H. & Zandieh, A. A smeared stiffener model for vibration analysis of anisogrid-stiffened composite conical shells using differential quadrature method. J Braz. Soc. Mech. Sci. Eng. 46, 38 (2024). https://doi.org/10.1007/s40430-023-04606-y

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