**2. Modeled structure: geometry and operating conditions**

The composite beam studied in this research is a slender tubular structure with optimizing prepreg lay-up sequence. Its cross-section is comprised of two straight stripes and two half-circles, each of them is made of eight laminating layers (see **Figure 1**), that is similar to those studied in [19]. The simulated structure is a greatly simplified design of D-like spar of the helicopter main rotor blade. Such spars are typically characterized by thickening of the cross section near the root, decreasing thickness of the airfoil and its twist as it approaches the end of the blade, without tapering of airfoil. The CAD model of the structure was built by using NX CAD (Siemens®) capabilities and converted into the Structural Mechanics Comsol Multiphysics environment.

In order to define the structural anisotropy of the laminate and the distributed external loads, the curvilinear coordinate systems (**<sup>e</sup>**<sup>1</sup> , **e**2 , **<sup>e</sup>**3) have been intended on the external surface of the tube. To do this, we used a diffusion method which solves Laplace's equation Δ*U* = 0 and computes the vector field as −∇*U*. The boundary conditions were assigned as follows: on the internal surface *U*int <sup>=</sup> 1, *Uout* <sup>=</sup> 0 on the outer surface, and −**<sup>n</sup>** <sup>⋅</sup> <sup>∇</sup>*<sup>U</sup>* <sup>=</sup> 0 on the end faces. The normal to the surface is assigned as **e**<sup>1</sup> <sup>=</sup> <sup>−</sup>∇*U*/|−∇*U*|. The second component is the unit vector parallel to the beam longitudinal axis and third one **e**<sup>3</sup> is normal to the plane (**e**<sup>1</sup> , **<sup>e</sup>**2). The curvilinear coordinates are presented in **Figure 2**.

The root end face is fixed (*u*, *v*, *w*) = 0, where *u*, *v*, *w* are the displacements along the global axes (*x*, *y*, *z*), respectively. Three different external loads can be applied alternately or together:

$$F\_{\text{tors}} = \begin{cases} 0 \cdot \mathbf{e}\_1 \\ 0 \cdot \mathbf{e}\_2 \\ 10^4 \cdot H(\text{x} - 3, 1) \cdot \mathbf{e}\_3 \end{cases} \tag{1}$$

$$F\_{load} = \begin{cases} 0 \cdot \mathfrak{g} \\ 70 \cdot yload \cdot \mathfrak{n} \\ 45 \cdot zload \cdot \mathfrak{g} \end{cases} \tag{2}$$

In Eq. (1) the twisting load *H*(*x*, *x*) is the smoothed step-function with the coordinate of step *<sup>x</sup>* and smoothing width *x*. The bending force *Fbend* is defined by the components in the global coordinate system whose unit vectors are (, , ) and by the parameters *yload*, *zload*, which can

**Figure 1.** Sketch of the beam cross-sections (a), 3D view of the CAD model and magnifying view on the cross-section of

Optimization of Lay-Up Stacking for a Loaded-Carrying Slender Composite Beam

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39

take alternatively or together, have the values of 0 and (or) one.

optimizing structure.

Optimization of Lay-Up Stacking for a Loaded-Carrying Slender Composite Beam http://dx.doi.org/10.5772/intechopen.76566 39

beam experiencing distributed bending and torsion forces. Then we determine the elastic properties of laminates used in the modeled tube. We start from the mechanical properties of reinforcing fibers and epoxy resin, and then we determine the properties of the unidirectional lamina and, finally, the laminate properties. Our optimization approach contains three sequential stages. The preliminary stage is based on the consideration of the angular distribution of all engineering constants of laminates. This analysis allows us to choose the small enough set of "candidate" lay-ups, which should be used at the modeling of the mechanical response of the studied structure at three different load scenarios. The next "candidates" higher level "candidates"—are appointed for the final dynamic test, which includes applying full load to the preferred structures and gives us the possibility to make the expert decision about final choice of quasi-optimal structure. Last, we discuss some considerations influenc-

The composite beam studied in this research is a slender tubular structure with optimizing prepreg lay-up sequence. Its cross-section is comprised of two straight stripes and two half-circles, each of them is made of eight laminating layers (see **Figure 1**), that is similar to those studied in [19]. The simulated structure is a greatly simplified design of D-like spar of the helicopter main rotor blade. Such spars are typically characterized by thickening of the cross section near the root, decreasing thickness of the airfoil and its twist as it approaches the end of the blade, without tapering of airfoil. The CAD model of the structure was built by using NX CAD (Siemens®) capabilities and converted into the Structural Mechanics Comsol Multiphysics environment. In order to define the structural anisotropy of the laminate and the distributed external loads,

tube. To do this, we used a diffusion method which solves Laplace's equation Δ*U* = 0 and computes the vector field as −∇*U*. The boundary conditions were assigned as follows: on the internal surface *U*int <sup>=</sup> 1, *Uout* <sup>=</sup> 0 on the outer surface, and −**<sup>n</sup>** <sup>⋅</sup> <sup>∇</sup>*<sup>U</sup>* <sup>=</sup> 0 on the end faces. The normal to the surface is assigned as **e**<sup>1</sup> <sup>=</sup> <sup>−</sup>∇*U*/|−∇*U*|. The second component is the unit vector parallel to

The root end face is fixed (*u*, *v*, *w*) = 0, where *u*, *v*, *w* are the displacements along the global axes (*x*, *y*, *z*), respectively. Three different external loads can be applied alternately or together:

104 ⋅ *H*(*x* − 3, 1) ⋅ **e**<sup>3</sup>

70 ⋅ *yload* ⋅ 45 ⋅ *zload* ⋅

⎧ ⎪ ⎨ ⎪ ⎩

0 ⋅ **e**<sup>1</sup> 0 ⋅ **e**<sup>2</sup>

> ⎧ ⎪ ⎨ ⎪ ⎩

0 ⋅

is normal to the plane (**e**<sup>1</sup>

, **<sup>e</sup>**3) have been intended on the external surface of the

, **<sup>e</sup>**2). The curvilinear coor-

(1)

(2)

ing the final choice of the best lay-up parameters that are the design variables.

, **e**2

**2. Modeled structure: geometry and operating conditions**

the curvilinear coordinate systems (**<sup>e</sup>**<sup>1</sup>

38 Optimum Composite Structures

the beam longitudinal axis and third one **e**<sup>3</sup>

*Ftors* =

*Fbend* =

dinates are presented in **Figure 2**.

**Figure 1.** Sketch of the beam cross-sections (a), 3D view of the CAD model and magnifying view on the cross-section of optimizing structure.

In Eq. (1) the twisting load *H*(*x*, *x*) is the smoothed step-function with the coordinate of step *<sup>x</sup>* and smoothing width *x*. The bending force *Fbend* is defined by the components in the global coordinate system whose unit vectors are (, , ) and by the parameters *yload*, *zload*, which can take alternatively or together, have the values of 0 and (or) one.

**Figure 2.** Curvilinear local coordinate systems on the beam surface.

monitored by extensometers. During these experiments, the following elastic moduli have

Remark: Index 1 corresponds to the longitudinal ply orientation (along the fibers), whereas index 2 to the transversal

**Experiment Mixing rule FE modeling**

Optimization of Lay-Up Stacking for a Loaded-Carrying Slender Composite Beam

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**Components Young module, GPa Poisson ratio Shear module, GPa** Glass fibers 87.5 (longitudinal) *ν*<sup>12</sup> = *ν*<sup>13</sup> = 0.255 *G*<sup>12</sup> = *G*<sup>13</sup> = 2.594 Epoxy resin 2.55 0.364 0.94 (calculated)

, MPa **49,500 ± 2500 49,000 48,090**

, MPa 6000 ± 500 5500 **8042** ν<sup>12</sup> **0.315 ± 0.015 0.31 0.299** ν<sup>23</sup> NA NA **0.459** G23, MPa 2600 ± 200 2000 **2758** G12, MPa NA NA **2847**

**Table 1.** Mechanical properties of the composite constituents.

**Elastic module Determination method**

The properties of unidirectional lamina, which has transversely isotropic symmetry, were determined both numerically and experimentally for the samples with volume fraction Vf = 0.534. In order to determine the parameters inaccessible through the experimental tests, we used the mixing rule and FE modeling in Abaqus environment according to the recommendation of [20, 21]. The values all found properties are shown in **Table 2**, where bold numbers denote the most reliable values, which have been used in the follow-up investigation. These values have been used for determination of elastic moduli of laminates of chosen

In order to diminish the formation of the residual stress and strains, which can arise during cure we use only 8-layers symmetric balanced laminates in our investigation. All used

For each lay-up scheme the elastic moduli were determined independently by two methods: by the finite element method and on the base of the classical laminates theory. The last one

**4. Determination of elastic moduli for laminates**

schemes of lamina stacking are shown in **Figure 4** with their designations.

proposes the following relationships for the elastic moduli of laminate.

been obtained (see **Table 1**).

**Table 2.** Mechanical properties of lamina.

structures.

ply orientation.

E1

E2

**Figure 3.** FE mesh used at the mechanics problem.

The FE mesh has been constructed for the given geometry as swept mesh (see **Figure 3**) consisting of 128,893 domain elements, 37,959 boundary elements and 1350 edge elements. The number of degrees of freedom solved for is 625,409.
