# Cross-sectional Parameterization and Beam Properties#

The cross-section is comprised of the following components: box spar, front laminate, back laminate, front filling, back filling and non-structural mass, as shown in Fig. 3.

## Parameterization#

Parameters can be in general grouped into two sets, geometry related and layup related.

### Geometry#

In this example, four geometric parameters are defined in the non-dimensional frame where the chord length is 1, as shown in Fig. 4. The origin of this frame is placed at the leading edge. The size and location of the box spar are defined by the two parameters $$a_2^{wl}$$ and $$a_2^{wt}$$, which are in fact the horizontal coordinates of the leading and trailing webs, respectively. The non-structural mass is assumed to be a circular object in this example. Hence, the three parameters defining the size and location are the horizontal coordinate $$a_2^{nsm}$$ and vertical coordinate $$a_3^{nsm}$$ of the center, and the radius $$r^{nsm}$$. Figure 4 Parameters for the geometry in the non-dimensional frame, where the chord length is 1.#

### Layup#

Layup related parameters are used to define materials ($$l$$), fiber angles ($$\theta$$) and numbers of plies ($$n$$) for the laminates of box spar, cap and overwrap. In this example, the box spar laminate is assumed to have four layers. All four layers have the same material ($$l^s$$), but can have different fiber angles ($$\theta_1^s, \theta_2^s, \theta_3^s, \theta_4^s$$) and numbers of plies ($$n_1^s, n_2^s, n_3^s, n_4^s$$). The cap laminate is assumed to have a single layer with three parameters: $$l^c, \theta^c, n^c$$. The overwrap laminate is also assumed having a single layer with three parameters: $$l^o, \theta^o, n^o$$.

The overall layup design of the three lamiante components are listed in Table 4, Table 5 and Table 6.

Table 4 Layup of the box spar laminate#

Layer

Lamina

Fiber angle [deg]

Ply count

Base

T300 15k/976

0

2

1

$$l^s$$

$$\theta^s_1$$

$$n^s_1$$

2

$$l^s$$

$$\theta^s_2$$

$$n^s_2$$

3

$$l^s$$

$$\theta^s_3$$

$$n^s_3$$

4

$$l^s$$

$$\theta^s_4$$

$$n^s_4$$

Table 5 Layup of the front laminate#

Layer

Lamina

Fiber angle [deg]

Ply count

Cap

Aluminum 8009

0

2

Base

T300 15k/976

0

2

1

$$l^f$$

$$\theta^f$$

$$n^f$$

Table 6 Layup of the back laminate#

Layer

Lamina

Fiber angle [deg]

Ply count

Base

T300 15k/976

0

2

1

$$l^b$$

$$\theta^b$$

$$n^b$$

A summary of the parameters is listed in Table 7.

Table 7 Summary of parameters#

Symbol

Name in input files

Description

$$a_2^{wl}$$

wl_a2

Non-dimensional horizontal coordinate of the leading web of the box spar

$$a_2^{wt}$$

wt_a2

Non-dimensional horizontal coordinate of the trailing web of the box spar

$$a_2^{nsm}$$

pnsmc_a2

Non-dimensional horizontal coordinate of the center of the non-structural mass

$$a_3^{nsm}$$

pnsmc_a3

Non-dimensional vertical coordinate of the center of the non-structural mass

$$r^{nsm}$$

nsmr

Non-dimensional radius of the non-structural mass

$$l^s$$

mi_spar_1

Material (lamina) selection of the box spar layup

$$\theta_1^s$$

fo_spar_1

Fiber angle of layer 1 of the box spar layup

$$\theta_2^s$$

fo_spar_2

Fiber angle of layer 2 of the box spar layup

$$\theta_3^s$$

fo_spar_3

Fiber angle of layer 3 of the box spar layup

$$\theta_4^s$$

fo_spar_4

Fiber angle of layer 4 of the box spar layup

$$n_1^s$$

np_spar_1

Number of plies of layer 1 of the box spar layup

$$n_2^s$$

np_spar_2

Number of plies of layer 2 of the box spar layup

$$n_3^s$$

np_spar_3

Number of plies of layer 3 of the box spar layup

$$n_4^s$$

np_spar_4

Number of plies of layer 4 of the box spar layup

$$l^c$$

mi_le

Material (lamina) selection of the cap layup

$$\theta^c$$

fo_le

Fiber angle of the cap layup

$$n^c$$

np_le

Number of plies of the cap layup

$$l^o$$

mi_te

Material (lamina) selection of the overwrap layup

$$\theta^o$$

fo_te

Fiber angle of the overwrap layup

$$n^o$$

np_te

Number of plies of the overwrap layup

Some other parameters used are listed below:

Table 8 Summary of other parameters#

Name in input files

Description

chord

Chord length of the cross-section

oa2

Non-dimensional horizontal coordinate of the origin of the frame $$\mathbf{x}$$ used in VABS.

pfte2_a2

Non-dimensional location of point marking the coarse meshes in the filling region

mesh_size

Global mesh size

mesh_size_fill

Mesh size for the filling regions

## Materials#

Materials used in this example are summarized in Table 9. Lamina thickness is given in Table 10. ‘Aluminum 80009’ is used as the outermost layer of the cap. ‘Lead’ is used for the non-structural mass. ‘Rohacell 70’ is used at the leading filling region. ‘Plascore PN2-3/16OX3.0’ is used at the trailing filling region. ‘T300 15k/976’ is used as the base layer for the spar, cap and overwrap, to prevent zero layers during the optimization. The last four rows in the table are the candidate materials for the layup design of the three components.

Table 9 Material properties#

Material

Density

$$E_1$$

$$E_2$$

$$E_3$$

$$\nu_{12}$$

$$\nu_{13}$$

$$\nu_{23}$$

$$G_{12}$$

$$G_{13}$$

$$G_{23}$$

[$$\mathrm{lbf\cdot s^2/in^4}$$]

[$$\mathrm{psi}$$]

[$$\mathrm{psi}$$]

[$$\mathrm{psi}$$]

[$$\mathrm{psi}$$]

[$$\mathrm{psi}$$]

[$$\mathrm{psi}$$]

Aluminum 8009

$$0.271959\times 10^{-3}$$

$$13.1\times 10^6$$

$$0.33$$

$$1.060957\times 10^{-3}$$

$$1.0\times 10^{-3}$$

$$0$$

Rohacell 70

$$7.040895\times 10^{-6}$$

$$13.125\times 10^3$$

$$13.125\times 10^3$$

$$13.125\times 10^3$$

$$0.315$$

$$0.315$$

$$0.300$$

$$4.118\times 10^3$$

$$4.118\times 10^3$$

$$4.118\times 10^3$$

Plascore PN2-3/16OX3.0

$$4.509066\times 10^{-6}$$

$$1.0\times 10^3$$

$$1.0\times 10^3$$

$$20\times 10^3$$

$$0.30$$

$$0.01$$

$$0.01$$

$$1.0\times 10^3$$

$$3.5\times 10^3$$

$$5.799\times 10^3$$

T300 15k/976

$$0.149716\times 10^{-3}$$

$$19.6\times 10^6$$

$$1.34\times 10^6$$

$$0.348$$

$$0.91\times 10^6$$

AS4 12k/E7K8

$$0.145973\times 10^{-3}$$

$$19.3\times 10^6$$

$$1.23\times 10^6$$

$$0.32$$

$$7.31\times 10^6$$

S2/SP381

$$0.173109\times 10^{-3}$$

$$7.05\times 10^6$$

$$1.97\times 10^6$$

$$0.263$$

$$0.59\times 10^6$$

T650-35 12k/976

$$0.148781\times 10^{-3}$$

$$22.0\times 10^6$$

$$1.30\times 10^6$$

$$0.115$$

$$0.745\times 10^6$$

T700 24K/E765

$$0.145037\times 10^{-3}$$

$$18.71\times 10^6$$

$$1.36\times 10^6$$

$$0.319$$

$$0.65\times 10^6$$

Table 10 Lamina thickness#

Material

Thickness [$$in$$]

Aluminum 8009

0.01

T300 15k/976

0.0053

AS4 12k/E7K8

0.0054

S2/SP381

0.0092

T650-35 12k/976

0.0052

T700 24K/E765

0.0056

## Properties#

The beam properties calculated by VABS listed in Table 11 will be needed in the optimization. Fig. 5 shows the points and axes with respect to which some properties are calculated. The two bending stiffnesses are given with respect to the principal bending axes. The horizontal locations of the shear and mass centers are given with respect to the modeling origin, which is the quarder chord in this example.

A summary of the properties is listed in Table 11.

Table 11 Summary of properties#

Symbol

Name in input files

Description

$$m$$

mu

Mass per unit length

$$GJ$$

gj

Torsional stiffness

$$EI_f$$

ei22

Flapping bending stiffness

$$EI_l$$

ei33

$$SC_2$$
sc2
$$MC_2$$
mc2