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

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

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 proposes the following relationships for the elastic moduli of laminate.

$$\begin{aligned} \overline{E}\_{\times} &= \frac{A\_{11}A\_{22} - A\_{12}^2}{A\_{22}H} \; ; \quad \overline{E}\_{\times} = \frac{A\_{11}A\_{22} - A\_{12}^2}{A\_{11}H} \; ; \\\overline{G}\_{\times y} &= \frac{A\_{66}}{H} \; ; \quad \qquad \overline{v}\_{\times y} = \frac{A\_{12}}{A\_{22}} .\end{aligned} \tag{3}$$

where indices *x*, *y* correspond to the longitudinal and transversal directions of laminate, respectively, and the elements of {*A*} matrix are calculated by the formula [22, 23].

$$A\_{ij} = \sum\_{k=1}^{N} \overline{Q}\_{ijk} (\mathbf{z}\_k - \mathbf{z}\_{k-1}) \tag{4}$$

accurate identification of the modules described in [24]. All modules to be determined in the numerical experiments are averaged by integration along the lines where the pure tensile or

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

Geometry of the FE models consists of eight separated layers, each of which is characterized by its orientation of lamina. Numerical calculation of elastic modules has been

**Figure 6.** Some examples of longitudinal, transverse elastic moduli and Poisson ratios dependencies on the lay-up angles

for the laminate stacking I, II, III, IV (see **Figure 4**).

of lamina orientation.

43

http://dx.doi.org/10.5772/intechopen.76566

implemented for all lay-ups (see **Figure 4**) with the step Δ*φ* = 100

shear occurs (see **Figure 5**).

Where, *Q*¯*ijk* are the *i*, *<sup>j</sup>*-th elements of the reduced stiffness matrix *Q*¯ for the *k*-th layer, *zk*−<sup>1</sup> , *zk* are the bounds of the *k*-th layer and *H* is the laminate thickness.

The FE model for determining of elastic moduli simulated the experimental tests according to the standards ASTM D 3039–95 and ASTM D 5379–93 with the numerical methods for the

**Figure 4.** Used schemes of lamina stacking.

**Figure 5.** Geometry (a, c) and postprocessing results (b, d) of models simulating the mechanical tests ASTM D 3039–95 "Test Method for Tensile Properties of Polymer Matrix Composite Materials" (a, b) and ASTM D 5379–93 "Test Method for Shear Properties of Composite Materials by V-Notched Beam Method" (c, d).

accurate identification of the modules described in [24]. All modules to be determined in the numerical experiments are averaged by integration along the lines where the pure tensile or shear occurs (see **Figure 5**).

42 Optimum Composite Structures

*<sup>E</sup>*¯*<sup>x</sup>* <sup>=</sup> *<sup>A</sup>*<sup>11</sup> *<sup>A</sup>*<sup>22</sup> <sup>−</sup> *<sup>A</sup>*<sup>12</sup>

*<sup>G</sup>*¯*xy* <sup>=</sup> \_\_\_ *A*<sup>66</sup>

the bounds of the *k*-th layer and *H* is the laminate thickness.

*Aij* = ∑

**Figure 4.** Used schemes of lamina stacking.

2 \_\_\_\_\_\_\_\_\_

*<sup>H</sup>* ; *ν*

respectively, and the elements of {*A*} matrix are calculated by the formula [22, 23].

*k*=1 *N*

Where, *Q*¯*ijk* are the *i*, *<sup>j</sup>*-th elements of the reduced stiffness matrix *Q*¯ for the *k*-th layer, *zk*−<sup>1</sup>

The FE model for determining of elastic moduli simulated the experimental tests according to the standards ASTM D 3039–95 and ASTM D 5379–93 with the numerical methods for the

**Figure 5.** Geometry (a, c) and postprocessing results (b, d) of models simulating the mechanical tests ASTM D 3039–95 "Test Method for Tensile Properties of Polymer Matrix Composite Materials" (a, b) and ASTM D 5379–93 "Test Method

for Shear Properties of Composite Materials by V-Notched Beam Method" (c, d).

*<sup>A</sup>*<sup>22</sup> *<sup>H</sup>* ; *<sup>E</sup>*¯*<sup>y</sup>* <sup>=</sup> *<sup>A</sup>*<sup>11</sup> *<sup>A</sup>*<sup>22</sup> <sup>−</sup> *<sup>A</sup>*<sup>12</sup>

where indices *x*, *y* correspond to the longitudinal and transversal directions of laminate,

¯*xy* <sup>=</sup> \_\_\_ *A*<sup>12</sup> *A*<sup>22</sup> .

2 \_\_\_\_\_\_\_\_\_ *<sup>A</sup>*<sup>11</sup> *<sup>H</sup>* ;

*Q*¯*ijk*(*zk* − *zk*−1) (4)

(3)

, *zk* are Geometry of the FE models consists of eight separated layers, each of which is characterized by its orientation of lamina. Numerical calculation of elastic modules has been implemented for all lay-ups (see **Figure 4**) with the step Δ*φ* = 100 of lamina orientation.

**Figure 6.** Some examples of longitudinal, transverse elastic moduli and Poisson ratios dependencies on the lay-up angles for the laminate stacking I, II, III, IV (see **Figure 4**).

Both moduli calculation methods have demonstrated a good correspondence, but FE method provides higher accuracy at the calculation of the Poisson ratios and in-plane shear module *Gxy*. Some examples of obtained results are shown in **Figures 6** and **7**. Each point on these plots at the chosen angle *φ* (or *φ* and *ψ* for the scheme V) represents the

**Figure 7.** The in-plane shear module dependencies on the lay-up angles for the laminate stacking I, II, III, IV and V (see **Figure 4**).

**Figure 8.** Angular dependencies of the longitudinal, shear modules and in-plane Poisson ratio for the lay-ups II (left)

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

http://dx.doi.org/10.5772/intechopen.76566

45

and IV (right) (see **Figure 4**).

Optimization of Lay-Up Stacking for a Loaded-Carrying Slender Composite Beam http://dx.doi.org/10.5772/intechopen.76566 45

Both moduli calculation methods have demonstrated a good correspondence, but FE method provides higher accuracy at the calculation of the Poisson ratios and in-plane shear module *Gxy*. Some examples of obtained results are shown in **Figures 6** and **7**. Each point on these plots at the chosen angle *φ* (or *φ* and *ψ* for the scheme V) represents the

**Figure 7.** The in-plane shear module dependencies on the lay-up angles for the laminate stacking I, II, III, IV and V (see

**Figure 4**).

44 Optimum Composite Structures

**Figure 8.** Angular dependencies of the longitudinal, shear modules and in-plane Poisson ratio for the lay-ups II (left) and IV (right) (see **Figure 4**).

values of effective elastic moduli for the laminate with given lay-up. Note that according to the homogenization hypothesis this laminate is considered as the solid body with uniform structure.

In the next stage of investigation, the angular dependencies of the effective moduli were investigated. In order to avoid the significant reduction of the structural rigidity in some directions, these dependencies should not have sharp drops, corresponding to these directions. Some examples of such angular dependencies are shown in **Figure 8**. These dependencies have been used for the preliminary selections of the "candidates" lay-ups, which have been further used to test them in mechanical testing of the optimized slender structure.
