A new method for concurrent multi-scale design optimization of fiber-reinforced composite frames with fundamental frequency constraints

Abstract

This paper proposes an efficient methodology for concurrent multi-scale design optimization of composite frames considering specific design constraints to obtain the minimum structure cost when the fundamental frequency is considered as a constraint. To overcome the challenge posed by the strongly singular optimum and the weakness of the conventional polynomial material interpolation (PLMP) scheme, a new area/moment of inertia–density interpolation scheme, which is labeled as adapted PLMP (APLMP) is proposed. The APLMP scheme and discrete material optimization approach are employed to optimize the macroscopic topology of a frame structure and microscopic composite material selection concurrently. The corresponding optimization formulation and solution procedures are also developed and validated through numerical examples. Numerical examples show that the proposed APLMP scheme can effectively solve the singular optimum problem in the multi-scale design optimization of composite frames with fundamental frequency constraints. The proposed multi-scale optimization model for obtaining the minimum cost of structures with a fundamental frequency constraint is expected to provide a new choice for the design of composite frames in engineering applications.

Appendix Layer-wise constant shear beam theory

The transformed stress–strain relation of an orthotropic single lamina under the assumption of plane stress in the x–y plane without the transverse normal stress component in the structure coordinates [$x$ $y$ $z$] (see Fig. 21) can be written as

$$\left(\begin{array}{c}\begin{array}{c}{\sigma }_{x}\\ {\sigma }_{y}\\ {\sigma }_{xy}\end{array}\\ {\sigma }_{yz}\\ {\sigma }_{xz}\end{array}\right)=\left(\begin{array}{c}\begin{array}{c}{\bar{Q} }_{11}\\ {\bar{Q} }_{12}\\ {\bar{Q} }_{16}\end{array}\\ 0\\ 0\end{array}\begin{array}{c}\begin{array}{c}{\bar{Q} }_{12}\\ {\bar{Q} }_{22}\\ {\bar{Q} }_{26}\end{array}\\ 0\\ 0\end{array}\begin{array}{c}\begin{array}{c}{\bar{Q} }_{16}\\ {\bar{Q} }_{26}\\ {\bar{Q} }_{66}\end{array}\\ 0\\ 0\end{array}\begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ {\bar{Q} }_{44}\\ {\bar{Q} }_{45}\end{array}\begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ {\bar{Q} }_{45}\\ {\bar{Q} }_{55}\end{array}\right)\left(\begin{array}{c}\begin{array}{c}{\varepsilon }_{x}\\ {\varepsilon }_{y}\\ {\gamma }_{xy}\end{array}\\ {\gamma }_{yz}\\ {\gamma }_{xz}\end{array}\right)$$

(27)

where ${\bar{Q} }_{pq}$ ($p,q\in$ 1, 2, 4, 5, 6) is the transformed reduced stiffness. The reduced stiffness ${Q}_{pq}$ can be expressed as follows:

$${Q}_{11}=\frac{{E}_{11}}{1-{\nu }_{12}{\nu }_{21}}, {Q}_{22}=\frac{{E}_{22}}{1-{\nu }_{12}{\nu }_{21}}, {Q}_{12}=\frac{{{\nu }_{21}E}_{11}}{1-{\nu }_{12}{\nu }_{21}}=\frac{{{\nu }_{12}E}_{22}}{1-{\nu }_{12}{\nu }_{21}}$$

(28a)

$${Q}_{44}={G}_{23}, {Q}_{55}={G}_{13}, {Q}_{66}={G}_{12}$$

(28b)

Fig. 21

Schematic of the fiber winding angle of a composite beam with a circular cross-section

The invariant parameters ${U}_{1}-{U}_{6}$ are defined to efficiently calculate ${\bar{Q} }_{pq}$ as.

$${U}_{1}=\left(3{Q}_{11}+3{Q}_{22}+2{Q}_{12}+4{Q}_{66}\right)/8$$

(29a)

$${U}_{2}=\left({Q}_{11}-{Q}_{22}\right)/2$$

(29b)

$${U}_{3}=\left({Q}_{11}+{Q}_{22}-2{Q}_{12}-4{Q}_{66}\right)/8$$

(29c)

$${U}_{4}=\left({Q}_{11}+{Q}_{22}+6{Q}_{12}-4{Q}_{66}\right)/8$$

(29d)

$${U}_{5}=\left({Q}_{44}+{Q}_{55}\right)/2$$

(29e)

$${U}_{6}=\left({Q}_{44}-{Q}_{55}\right)/2$$

(29f)

Then, ${\bar{Q} }_{pq}$ can be expressed as.

$$\left\{\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}{\bar{Q} }_{11}\\ {\bar{Q} }_{22}\\ {\bar{Q} }_{12}\end{array}\\ {\bar{Q} }_{44}\end{array}\\ {\bar{Q} }_{55}\end{array}\\ {\bar{Q} }_{45}\\ {\bar{Q} }_{66}\end{array}\\ {\bar{Q} }_{16}\end{array}\\ {\bar{Q} }_{26}\end{array}\right\}=\left[\begin{array}{ccccc}{U}_{1}& {U}_{2}& 0& {U}_{3}& 0\\ {U}_{1}& {-U}_{2}& 0& {U}_{3}& 0\\ {U}_{4}& 0& 0& {-U}_{3}& 0\\ {U}_{5}& {U}_{6}& 0& 0& 0\\ {U}_{5}& {-U}_{6}& 0& 0& 0\\ 0& 0& {-U}_{6}& 0& 0\\ \frac{1}{2}({U}_{1}{-U}_{4})& 0& 0& {-U}_{3}& 0\\ 0& 0& \frac{1}{2}{U}_{2}& 0& {U}_{3}\\ 0& 0& \frac{1}{2}{U}_{2}& 0& {-U}_{3}\end{array}\right]\left\{\begin{array}{c}1\\ {\text{cos}}\left(2{\theta }_{i,j}\right)\\ {\text{sin}}\left(2{\theta }_{i,j}\right)\\ {\text{cos}}\left(4{\theta }_{i,j}\right)\\ {\text{sin}}\left(4{\theta }_{i,j}\right)\end{array}\right\}$$

(30)

where ${\theta }_{i,j}$ is the fiber winding angle for layer $j$ of tube $i$. The layers are numbered with the inner layer as the first layer. The schematic of the fiber winding angle definition is shown in Fig. 21 for a single layer, where $+{\theta }_{i,j}$ denotes the positive fiber winding angle; $x$, $y$, and $z$ are the beam structure coordinates; and 1, 2, and 3 are the principal material coordinates.

Assuming that the laminated beam is a one-dimensional component, the coordinate $y$ is the circumferential direction along a circular cross-section beam; thus, ${\sigma }_{y}={\sigma }_{yz}={\sigma }_{xy}=0$ (Jones 2014) is applied in Eq. (27), which yields

$${\sigma }_{x}={{E}_{x}}^{i,j}{\varepsilon }_{x} {\sigma }_{xz}={{G}_{xz}}^{i,j}{\gamma }_{xz}$$

(31)

where the equivalent elastic modulus along the x-direction and shear modulus in the x–z plane of the j-th layer of the i-th tube, ${{E}_{x}}^{i,j}$ and ${{G}_{xz}}^{i,j},$ are given respectively by

$${{E}_{x}}^{i,j}={\bar{Q} }_{11}+{\bar{Q} }_{12}\frac{({\bar{Q} }_{16}{\bar{Q} }_{26}-{\bar{Q} }_{12}{\bar{Q} }_{66})}{({\bar{Q} }_{22}{\bar{Q} }_{66}-{\bar{Q} }_{26}{\bar{Q} }_{26})}+{\bar{Q} }_{16}\frac{({\bar{Q} }_{12}{\bar{Q} }_{26}-{\bar{Q} }_{16}{\bar{Q} }_{22})}{({\bar{Q} }_{22}{\bar{Q} }_{66}-{\bar{Q} }_{26}{\bar{Q} }_{26})}$$

(32)

and

$${{G}_{xz}}^{i,j}=\left({\bar{Q} }_{55}-\frac{{{\bar{Q} }_{45}}^{2}}{{\bar{Q} }_{44}}\right),$$

(33)

respectively. With the derivation above, ${{E}_{x}}^{i,j}$ and ${{G}_{xz}}^{i,j}$ are expressed as a function of the fiber winding angles with fixed orthotropic material properties.