3. Mathematical model and numerical analysis of reinforced beam deformation

Three-point bending flexural test has been one of the standard techniques to determine physical and mechanical characteristics of materials. Figure 1 shows a scheme of physical model of threepoint bending of a beam with the rectangular cross section b � 2h, and the span l between the supports. The left edge of the beam is hinged, while the right one is supported freely. The force P is applied to the center of the beam. The model neglects the shape of the supports and assumes the occurring load P and support reactions RA and RB to be concentrated. In addition, the model neglects the possible heterogeneity of the deformations in the direction normal both to the longitudinal direction and to the load direction.

Figure 1. Three-point bending of a rectangular-sectioned beam.

In this case, the beam's upper part undergoes compression strain in the longitudinal direction, bottom part—tension strain. VSE-1212 polymer matrix and VKU-28 (T-800 carbon yarn plus VSE-1212 epoxy matrix) structured CFRP react differently to tension and compression. VKU-28 has been one of the most promising types of CFRPs that is going to be used in the latest generations of aircrafts. The effect of accounting for this factor on the computational results is essential. Further, these results are compared to acquired ones.

Estimations for G obtained using variational method are also obtained, and it is shown that

Gf <sup>þ</sup> <sup>ω</sup>mGm

Gm (4)

, A<sup>1212</sup> ¼ G: (5)

Gm <sup>þ</sup> <sup>ω</sup>mGf

gives more accurate approximation than (Eq. (2)) does. Hereinafter share moduli of matrix and

Components of effective stiffness tensor for unidirectionally reinforced layer in case of state of

, A<sup>1122</sup> <sup>¼</sup> <sup>ν</sup>21E<sup>1</sup>

Unwritten expressions can be obtained using symmetry rule or vanish. Hereinafter, we assume

The coefficients in the relations (Eq. (1)) for example are defined by the formulas given in [1, 17, 18].

Three-point bending flexural test has been one of the standard techniques to determine physical and mechanical characteristics of materials. Figure 1 shows a scheme of physical model of threepoint bending of a beam with the rectangular cross section b � 2h, and the span l between the supports. The left edge of the beam is hinged, while the right one is supported freely. The force P is applied to the center of the beam. The model neglects the shape of the supports and assumes the occurring load P and support reactions RA and RB to be concentrated. In addition, the model neglects the possible heterogeneity of the deformations in the direction normal both to the

3. Mathematical model and numerical analysis of reinforced beam

, Gf <sup>¼</sup> Ef

1 � ν12ν<sup>21</sup>

2 1 þ ν<sup>f</sup> :

<sup>G</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>ω</sup><sup>f</sup>

Gm <sup>¼</sup> Em

1 � ν12ν<sup>21</sup>

1 þ ω<sup>f</sup>

2 1ð Þ þ ν<sup>m</sup>

lower boundary

16 Optimum Composite Structures

fibers are

plane stress have the following form:

α, β ¼ 1, 2 and α 6¼ β.

deformation

<sup>A</sup>αααα <sup>¼</sup> <sup>E</sup><sup>α</sup>

longitudinal direction and to the load direction.

Figure 1. Three-point bending of a rectangular-sectioned beam.

Due to very low deformation rates, the classical theory of beam bending can be regarded as satisfactory for description of the equilibrium state. To this end, it is convenient to consider the beam's median surface as a reference one.

The beam's stress-strain state is characterized by the following values determined on the reference surface: the shear force Q xð Þ, the bending moment M xð Þ, the longitudinal force N xð Þ, and by the longitudinal displacement and bend (u xð Þ, w xð Þ respectively). The corresponding equilibrium equations are written as follows:

$$
\frac{dN}{d\mathbf{x}} = \mathbf{0}, \qquad \frac{d\mathbf{Q}}{d\mathbf{x}} = \mathbf{0}, \qquad \frac{dM}{d\mathbf{x}} = \mathbf{Q}.\tag{6}
$$

The reactions RA and RB can be determined by considering force equilibrium RA ¼ RB ¼ P=2. The bending moments at the support points are equal to zero: MA ¼ MB ¼ 0. The solution of the equation system (Eq. (6)) can be expressed as follows:

$$N=0, \quad Q(\mathbf{x}) = \begin{cases} P/2, & 0 \le \mathbf{x} \le \mathbf{l}/2, \\ -P/2, & \mathbf{l}/2 \le \mathbf{x} \le \mathbf{l}, \end{cases} \qquad M(\mathbf{x}) = \begin{cases} P\mathbf{x}/2, & 0 \le \mathbf{x} \le \mathbf{l}/2, \\ -P(\mathbf{x}-\mathbf{l})/2, & \mathbf{l}/2 \le \mathbf{x} \le \mathbf{l}. \end{cases} \tag{7}$$

Strain distribution for the beam's thickness can be obtained from the Kirchhoff-Love kinematic hypotheses:

$$
\varepsilon(\mathbf{x}, z) = e(\mathbf{x}) + z\kappa(\mathbf{x}) \tag{8}
$$

$$
\varepsilon(\mathbf{x}) = \frac{d\mathbf{u}}{d\mathbf{x}'} \qquad \qquad \kappa(\mathbf{x}) = -\frac{d^2 w}{d\mathbf{x}^2} \,. \tag{9}
$$

where εð Þ x; z is the strain in the beam; e xð Þ is the median surface strain; and κð Þx denotes changes in the median surface curvature. As mentioned earlier, the beam undergoes tension and compression strain, whose interface will be marked as z1. In this case, for the section area �h ≤ z ≤ z1, the strain will be negative, and for z<sup>1</sup> ≤ z ≤ h positive. At the interface of these two states, the strains ε vanish, so the interface itself is determined as follows:

$$z\_1 = -\frac{e}{\kappa}, \qquad -h \le z\_1 \le h. \tag{10}$$

The constitutive equation can be expressed as:

$$
\sigma^{\pm}(\mathbf{x}, \mathbf{z}) = f\_i^{\pm}(\mathbf{z}), \tag{11}
$$

where the superscript "+" refers to the areas with positive strains and "–" – to the area with negative ones; fð Þε denotes the approximation selected for the stress-strain curve (a linear function, a polynomial, or a combination of linear and power-law functions).

The longitudinal force N and the bending moment M in the beam cross section are determined by the equations:

$$\begin{aligned} N &= b \left( \int\_{-h}^{z\_1} \sigma^- dz + \int\_{z\_1}^h \sigma^+ dz \right), \\ M &= b \left( \int\_{-h}^{z\_1} \sigma^- z dz + \int\_{z\_1}^h \sigma^+ z dz \right). \end{aligned} \tag{12}$$

As the initial approximation at small values of the load P, the solutions obtained for the linear constitutive equations (Eq. (15)) were used. Since the computation is performed with a relatively small increment of P, in case of big values of P, one can use the computation results

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Having determined the change of median surface curvature from the equations (Eq. (13)) and

one can write down a differential equation to determine the beam bend. For that purpose, the

w xð Þ¼ <sup>w</sup>1ð Þ<sup>x</sup> , at <sup>x</sup> <sup>∈</sup>½ Þ <sup>0</sup>; <sup>l</sup>=<sup>2</sup> ,

Using the equation (Eq. (9)) and the beam's fixing conditions, a system of equations can be

w2 dx<sup>2</sup> ¼ �κ2, w1ð Þ¼ 0 w2ðÞ¼ l 0, w1ð Þ¼ l=2 w2ð Þ l=2 ,

dx :

The solution of these equations can be obtained using the methods of solving boundary-value problems for systems of ordinary differential equations. For that purpose, the modified collo-

Numerical analysis of deformation processes in VSE-1212 polymer matrix and VKU-28 structured carbon fiber is based on approximation of stress-strain curves and the three-point

2. specimen 2—VKU-28 structured carbon fiber (the specimen was cut out along the rein-

3. specimen 3—VKU-28 structured carbon fiber (the specimen was cut out perpendicular to

In Figure 2, one can see the simulation results for beam three-point bending, obtained through different approaches to approximation of the constitutive equations, and their comparison

Applying the linear dependencies to tension and compression has not resulted in adequate approximation even for 30% of curve. Using more complex than quadratic approximation laws

Further, three different specimens with the geometrical sizes l � 2h � b are considered:

κ1ð Þx , at x∈ ½ Þ 0; l=2 , κ2ð Þx , at x∈ ½ � l=2; l ,

w2ð Þx , at x ∈½ � l=2; l :

acquired at a previous step as the initial approximation for current step.

κð Þ¼ x

d2 w1

cation and least-residuals method [19–21] were applied.

forcement), 90 � 3:45 � 9:85 mm;

with the experimental data.

the reinforcement), 90 � 3:40 � 9:95 mm.

dw1ð Þ l=2

1. specimen 1—VSE-1212 polymer matrix, 75 � 4:78 � 10:05 mm,

bend function is expressed as follows:

dx<sup>2</sup> ¼ �κ1, <sup>d</sup><sup>2</sup>

dx <sup>¼</sup> dw2ð Þ <sup>l</sup>=<sup>2</sup>

(Eq. (14))

derived:

bending model.

Having substituted (Eq. (12)) with the relations (Eq. (8)), (Eq. (10)), (Eq. (11)) into and integrated it over the beam thickness, one obtains a system of equations to determine κ and e:

at 0 ≤ x ≤ l=2

$$\begin{cases} N(\kappa, e, \mathbf{x}) = 0, \\ M(\kappa, e, \mathbf{x}) = P\mathbf{x}/2, \end{cases} \tag{13}$$

at l=2 < x ≤ l

$$\begin{cases} N(\mathbf{x}, e, \mathbf{x}) = 0, \\ M(\mathbf{x}, e, \mathbf{x}) = -P(\mathbf{x} - \mathbf{l})/2. \end{cases} \tag{14}$$

The system of equations (Eq. (13)) and (Eq. (14)) in general case is nonlinear, but in the case of piecewise linear constitutive equations which take into account different strength and stiffness behavior in tension and compression expressed as follows:

$$
\sigma^{\pm}(\mathbf{x}, \mathbf{z}) = \mathbf{E}^{\pm}\mathbf{e},\tag{15}
$$

it can be solved analytically. In the nonlinear case, the Newton method is applied to solve the equations (Eq. (13)) and (Eq. (14)), and then, the linearized system

$$\begin{split} &N(\varepsilon\_{0},\kappa\_{0}) + \frac{\partial N(\varepsilon\_{0},\kappa\_{0})}{\partial \varepsilon}(\varepsilon-\varepsilon\_{0}) + \frac{\partial N(\varepsilon\_{0},\kappa\_{0})}{\partial \kappa}(\kappa-\kappa\_{0}) = 0, \\ &M(\varepsilon\_{0},\kappa\_{0}) + \frac{\partial M(\varepsilon\_{0},\kappa\_{0})}{\partial \varepsilon} \left(\varepsilon-\varepsilon\_{0}\right) + \frac{\partial M(\varepsilon\_{0},\kappa\_{0})}{\partial \kappa}(\kappa-\kappa\_{0}) = M(\mathbf{x}). \end{split}$$

can be solved for unknown values

$$\kappa = F(\varepsilon\_0, \kappa\_0, M(\mathbf{x})), \quad \varepsilon = G(\varepsilon\_0, \kappa\_0), \tag{16}$$

where ε<sup>0</sup> and κ<sup>0</sup> are the initial approximations, and M xð Þ is determined from (Eq. (7)).

As the initial approximation at small values of the load P, the solutions obtained for the linear constitutive equations (Eq. (15)) were used. Since the computation is performed with a relatively small increment of P, in case of big values of P, one can use the computation results acquired at a previous step as the initial approximation for current step.

where the superscript "+" refers to the areas with positive strains and "–" – to the area with negative ones; fð Þε denotes the approximation selected for the stress-strain curve (a linear function, a polynomial, or a combination of linear and power-law functions).

The longitudinal force N and the bending moment M in the beam cross section are determined

σ�dz þ

σ�zdz þ

Having substituted (Eq. (12)) with the relations (Eq. (8)), (Eq. (10)), (Eq. (11)) into and integrated it over the beam thickness, one obtains a system of equations to determine κ

> Nð Þ¼ κ;e; x 0, Mð Þ¼ κ;e; x Px=2,

Mð Þ¼� κ;e; x P xð Þ � l =2:

The system of equations (Eq. (13)) and (Eq. (14)) in general case is nonlinear, but in the case of piecewise linear constitutive equations which take into account different strength

it can be solved analytically. In the nonlinear case, the Newton method is applied to solve the

∂Nð Þ ε0; κ<sup>0</sup>

∂Mð Þ ε0; κ<sup>0</sup>

Nð Þ¼ κ;e; x 0,

ð h

σþdz

σþzdz

1 A,

> 1 A:

σ�ð Þ¼ x; z E�ε, (15)

<sup>∂</sup><sup>κ</sup> ð Þ¼ <sup>κ</sup> � <sup>κ</sup><sup>0</sup> <sup>0</sup>,

κ ¼ Fð Þ ε0; κ0; M xð Þ , ε ¼ Gð Þ ε0; κ<sup>0</sup> , (16)

<sup>∂</sup><sup>κ</sup> ð Þ¼ <sup>κ</sup> � <sup>κ</sup><sup>0</sup> M xð Þ

(12)

(13)

(14)

z1

ð h

z1

z ð1

0 @

�h

z ð1

0 @

�h

N ¼ b

M ¼ b

�

and stiffness behavior in tension and compression expressed as follows:

<sup>∂</sup><sup>ε</sup> ð Þþ <sup>ε</sup> � <sup>ε</sup><sup>0</sup>

<sup>∂</sup><sup>ε</sup> ð Þþ <sup>ε</sup> � <sup>ε</sup><sup>0</sup>

where ε<sup>0</sup> and κ<sup>0</sup> are the initial approximations, and M xð Þ is determined from (Eq. (7)).

�

equations (Eq. (13)) and (Eq. (14)), and then, the linearized system

∂Nð Þ ε0; κ<sup>0</sup>

∂Mð Þ ε0; κ<sup>0</sup>

Nð Þþ ε0; κ<sup>0</sup>

Mð Þþ ε0; κ<sup>0</sup>

can be solved for unknown values

by the equations:

18 Optimum Composite Structures

and e:

at 0 ≤ x ≤ l=2

at l=2 < x ≤ l

Having determined the change of median surface curvature from the equations (Eq. (13)) and (Eq. (14))

$$\kappa(\mathbf{x}) = \begin{cases} \kappa\_1(\mathbf{x}), & \text{at } \mathbf{x} \in [0, l/2), \\ \kappa\_2(\mathbf{x}), & \text{at } \mathbf{x} \in [l/2, l]. \end{cases}$$

one can write down a differential equation to determine the beam bend. For that purpose, the bend function is expressed as follows:

$$w(\mathbf{x}) = \begin{cases} w\_1(\mathbf{x}), & \text{at } \mathbf{x} \in [0, l/2), \\ w\_2(\mathbf{x}), & \text{at } \mathbf{x} \in [l/2, l]. \end{cases}$$

Using the equation (Eq. (9)) and the beam's fixing conditions, a system of equations can be derived:

$$\begin{aligned} \frac{d^2 w\_1}{d\mathbf{x}^2} &= -\kappa\_1, \quad \frac{d^2 w\_2}{d\mathbf{x}^2} = -\kappa\_2, \\ w\_1(0) &= w\_2(l) = 0, \quad w\_1(l/2) = w\_2(l/2), \\ \frac{dw\_1(l/2)}{d\mathbf{x}} &= \frac{dw\_2(l/2)}{d\mathbf{x}}. \end{aligned}$$

The solution of these equations can be obtained using the methods of solving boundary-value problems for systems of ordinary differential equations. For that purpose, the modified collocation and least-residuals method [19–21] were applied.

Numerical analysis of deformation processes in VSE-1212 polymer matrix and VKU-28 structured carbon fiber is based on approximation of stress-strain curves and the three-point bending model.

Further, three different specimens with the geometrical sizes l � 2h � b are considered:


In Figure 2, one can see the simulation results for beam three-point bending, obtained through different approaches to approximation of the constitutive equations, and their comparison with the experimental data.

Applying the linear dependencies to tension and compression has not resulted in adequate approximation even for 30% of curve. Using more complex than quadratic approximation laws

the approximations have shown the results close to experiment. At the same time for the test with the maximum load, the best option has still been application of quadratic approximations. Taking different strength and stiffness behavior in tension and compression into account has an essential effect. As it was demonstrated earlier, the tension tests of VKU-28 specimens produced nonlinear stress-strain curves, while the difference of characteristics between tension and compression reached 5–7% for the longitudinal reinforcements and 12–15%—for the

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For the polymer matrix this difference exceeded 15% (see Tables 3 and 4).

Approximation type Approximation coefficients MSD <sup>a</sup><sup>1</sup> � <sup>10</sup>�<sup>9</sup> <sup>a</sup><sup>2</sup> � <sup>10</sup>�<sup>9</sup> <sup>a</sup><sup>3</sup>

A1 160.8 1.4e – 2 A2 144.9 1.44e + 3 5.7e – 4 A3 144.0 1.66e + 3 –1.14e + 13 4.3e – 4 A4 143.0 8.87e + 2 1.87 4.1e – 4

A1 155.4 5.8e – 3 A2 160.2 –3.33e + 3 3.4e – 3 A3 155.9 4.31e + 3 �3.00e + 15 2.9e – 3 A4 157.7 �5.81e + 8 3.98 3.0e – 3

Approximation type Approximation coefficients MSD <sup>a</sup><sup>1</sup> � <sup>10</sup>�<sup>9</sup> <sup>a</sup><sup>2</sup> � <sup>10</sup>�<sup>9</sup> <sup>a</sup><sup>3</sup>

A1 7.37 1.1e – 2 A2 7.89 �9.22e + 1 6.4e – 4 A3 7.87 �8.97e + 1 2.54e + 11 4.3e – 4 A4 7.82 �2.23e + 2 2.20 4.8e – 4

A1 8.90 6.7e – 3 A2 9.21 �1.20e + 2 4.1e – 3 A3 8.96 1.16e + 2 �5.09e + 13 3.7e – 3 A4 9.04 �5.16e + 6 3.95 3.7e – 3

Table 2. Approximation coefficients for stress-strain curves of carbon fiber specimens reinforced in transverse direction

Table 1. Approximation coefficients for stress-strain curves of VKU-28 carbon fiber specimens reinforced in longitudinal

transverse ones (see Tables 1 and 2).

Tension of VKU-28 CFRP, ε∈½ � 0; 0:015

Compression of VKU-28 CFRP, ε∈½ � 0; 0:0018

direction and mean square deviation (MSD) of f xð Þ.

Tension of VKU-28 CFRP, ε∈½ � 0; 0:0076

Compression of VKU-28 CFRP, ε∈½ � 0; 0:0034

and mean square deviation (MSD) of f xð Þ.

Figure 2. Experimental (solid curves) and dependencies of beam-deflection and load obtained in simulation: linear approximation (1); quadratic approximation by a polynomial of the second degree (2); cubic approximation (3); linear and power-law approximation (4); a–c are specimens 1 3 respectively; d—the solution to a three-point bending problem without account for the different strength and stiffness behavior in tension (curve 1) and compression (curve 2). The solid line shows the results of mechanical tests.

at first led to a significant deviation from the experimental curve and then to divergence of the Newton method iteration process. This is explained by the fact that in tension tests, due to specimens' fragility, the strain range for the polymer matrix specimens was limited to 2%, while in the bending tests, the strains in tension zone reached 4–5%.

Thus, to solve the bending problem, the tension curve was extrapolated into the domain of high strains. The extrapolations obtained using a polynomial of the third degree, and by linear and power-law function reached the maximum too quickly and then started to decrease, which is against the physics behind the deformation process. A similar effect was observed when calculating the bending of the carbon-fiber specimens cut out along direction of reinforcement filler.

The calculations using quadratic approximation and extrapolation of tension curves and approximation of compression curves within a short (up to 6%) segment have turned out to be best for qualitative and quantitative description of the nonlinear character of VSE-1212 polymer matrix bending. In the case of the specimen cut out perpendicular to direction of its reinforcement, all the approximations have shown the results close to experiment. At the same time for the test with the maximum load, the best option has still been application of quadratic approximations.

Taking different strength and stiffness behavior in tension and compression into account has an essential effect. As it was demonstrated earlier, the tension tests of VKU-28 specimens produced nonlinear stress-strain curves, while the difference of characteristics between tension and compression reached 5–7% for the longitudinal reinforcements and 12–15%—for the transverse ones (see Tables 1 and 2).


For the polymer matrix this difference exceeded 15% (see Tables 3 and 4).

Table 1. Approximation coefficients for stress-strain curves of VKU-28 carbon fiber specimens reinforced in longitudinal direction and mean square deviation (MSD) of f xð Þ.


at first led to a significant deviation from the experimental curve and then to divergence of the Newton method iteration process. This is explained by the fact that in tension tests, due to specimens' fragility, the strain range for the polymer matrix specimens was limited to 2%,

Figure 2. Experimental (solid curves) and dependencies of beam-deflection and load obtained in simulation: linear approximation (1); quadratic approximation by a polynomial of the second degree (2); cubic approximation (3); linear and power-law approximation (4); a–c are specimens 1 3 respectively; d—the solution to a three-point bending problem without account for the different strength and stiffness behavior in tension (curve 1) and compression (curve 2). The solid

Thus, to solve the bending problem, the tension curve was extrapolated into the domain of high strains. The extrapolations obtained using a polynomial of the third degree, and by linear and power-law function reached the maximum too quickly and then started to decrease, which is against the physics behind the deformation process. A similar effect was observed when calculating the bending of the carbon-fiber specimens cut out along direction

The calculations using quadratic approximation and extrapolation of tension curves and approximation of compression curves within a short (up to 6%) segment have turned out to be best for qualitative and quantitative description of the nonlinear character of VSE-1212 polymer matrix bending. In the case of the specimen cut out perpendicular to direction of its reinforcement, all

while in the bending tests, the strains in tension zone reached 4–5%.

of reinforcement filler.

line shows the results of mechanical tests.

20 Optimum Composite Structures

Table 2. Approximation coefficients for stress-strain curves of carbon fiber specimens reinforced in transverse direction and mean square deviation (MSD) of f xð Þ.


Tables 1–4 show approximation results for the above-presented stress-strain curves by differ-

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23

However, if bending tests have been performed to determine an elasticity module of CFRPs, different strength and stiffness behavior in tension and compression is compensated and one

It is useful to consider the effect of the way for determining and setting of the mechanical characteristics on modeling of three-point bending of the carbon-fiber beam cutout perpendicular to its reinforcements. Figure 2d shows the solutions obtained while using a linear approximation of the constitutive equations with equal elastic moduli for tension and compression: for curve 1 the modulus was obtained from tension experiments, for curve 2—from compres-

As one can see the calculated linear results without account for the different strength and stiffness behavior in tension and compression have differed from the results of mechanical

Most of the real CFRP structures under day-to-day service conditions bear complex loads that result in formation of tension, compression and bending zones as well as their combinations in the structures. Applying the traditional methods for determination of material characteristics in combination with linear deformation models (in particular those that do not account for the different strength and stiffness behavior in tension and compression) for calculation of such structures, one risks to distort the deformation and stress pattern significantly, which, in its turn, results in either underestimation or overestimation of the structure's strength and rigidity. Keeping in mind that carbon fibers are used for manufacturing of high-duty structures, their computation demands different strength and stiffness behavior in tension and compression to be taken into account.

Composite overwrapped pressure vessels (COPV) are used in the rocket and spacecraft making industry due to their high strength and lightweight. Consisting of a thin, nonstructural liner wrapped with a structural fiber composite COPV are produced to hold the inner pressure of tens and hundreds atmospheres. COPV have been one of the most actual and perspective

Designing of a highly reliable and efficient COPV requires a technology for analysis of its deformation behavior and strength assessment. This technology should allow one to obtain

ent functions at different intervals:

obtains some averaged characteristic.

tests (the solid curve) by more than 15%.

sion ones (see Table 2).

1. by the linear approximation σ ¼ a1ε (A1),

2. by the polynomial of the second degree <sup>σ</sup> <sup>¼</sup> <sup>a</sup>1<sup>ε</sup> <sup>þ</sup> <sup>a</sup>2ε<sup>2</sup> (A2),

3. by the polynomial of the third degree <sup>σ</sup> <sup>¼</sup> <sup>a</sup>1<sup>ε</sup> <sup>þ</sup> <sup>a</sup>2ε<sup>2</sup> <sup>þ</sup> <sup>a</sup>3ε<sup>3</sup> (A3),

4. Numerical analysis and design of pressure vessels

directions of research, supported especially by NASA [22, 23].

4. by a combination of linear and power-law functions <sup>σ</sup> <sup>¼</sup> <sup>a</sup>1<sup>ε</sup> <sup>þ</sup> <sup>a</sup>2ε<sup>a</sup><sup>3</sup> (A4).

Table 3. Approximation coefficients for tension curves of VSE-1212 polymer matrix, ε∈½ � 0; 0:018 and mean square deviation (MSD) of f xð Þ.


Table 4. Approximation coefficients for compression curves of VSE-1212 polymer matrix and mean square deviation (MSD) of f xð Þ.

Tables 1–4 show approximation results for the above-presented stress-strain curves by different functions at different intervals:

1. by the linear approximation σ ¼ a1ε (A1),

Approximation type Approximation coefficients MSD <sup>a</sup><sup>1</sup> � <sup>10</sup>�<sup>9</sup> <sup>a</sup><sup>2</sup> � <sup>10</sup>�<sup>9</sup> <sup>a</sup><sup>3</sup>

A1 3.30 — — 2.9e � 2 A2 3.90 �4.38e + 1 — 1.5e � 3 A3 3.83 �3.17e + 1 �4.94e + 11 6.7e � 4 A4 3.80 �1.05e + 2 2.25 6.6e � 4

A1 3.33 2.7e � 2 A2 3.89 �4.02e + 1 1.8e � 3 A3 3.80 �2.48e + 1 �6.30e + 2 4.0e � 4 A4 3.77 �1.40e + 2 2.35 2.7e � 4

Table 3. Approximation coefficients for tension curves of VSE-1212 polymer matrix, ε∈½ � 0; 0:018 and mean square

Approximation type Approximation coefficients MSD

A1 0.77 2.3e � 1 A2 1.60 �3.97 8.2e � 2 A3 2.36 �1.29e + 1 2.37e + 10 1.4e � 2 A4 �5.71 5.49 0.90 3.8e � 2

A1 0.69 3.5e � 1 A2 1.69 �5.22 1.4e � 1 A3 2.71 �1.84e + 1 3.84e + 1 3.6e � 2 A4 �2.07 1.72 0.72 4.8e � 2

A1 2.10 7.2e � 2 A2 3.05 �2.12e + 1 4.2e � 3 A3 3.18 �2.85e + 1 9.13e + 1 1.1e � 3 A4 3.31 �1.24e + 1 1.75 1.9e � 3

Table 4. Approximation coefficients for compression curves of VSE-1212 polymer matrix and mean square deviation

Compression of VSE-1212 polymer matrix ε∈½ � 0; 0:06 (Variable cross section, shortened test area)

<sup>a</sup><sup>1</sup> � <sup>10</sup>�<sup>9</sup> <sup>a</sup><sup>2</sup> � <sup>10</sup>�<sup>9</sup> <sup>a</sup><sup>3</sup>

Strain of VSE-1212 polymer matrix (Constant cross section) ε∈½ � 0; 0:018

Strain of VSE-1212 polymer matrix (Variable cross section)ε∈½ � 0; 0:018

Compression of VSE-1212 polymer matrix (Constant cross section), ε∈½ � 0; 0:28

Compression of VSE-1212 polymer matrix (Variable cross section), ε∈½ � 0; 0:28

deviation (MSD) of f xð Þ.

22 Optimum Composite Structures

(MSD) of f xð Þ.


However, if bending tests have been performed to determine an elasticity module of CFRPs, different strength and stiffness behavior in tension and compression is compensated and one obtains some averaged characteristic.

It is useful to consider the effect of the way for determining and setting of the mechanical characteristics on modeling of three-point bending of the carbon-fiber beam cutout perpendicular to its reinforcements. Figure 2d shows the solutions obtained while using a linear approximation of the constitutive equations with equal elastic moduli for tension and compression: for curve 1 the modulus was obtained from tension experiments, for curve 2—from compression ones (see Table 2).

As one can see the calculated linear results without account for the different strength and stiffness behavior in tension and compression have differed from the results of mechanical tests (the solid curve) by more than 15%.

Most of the real CFRP structures under day-to-day service conditions bear complex loads that result in formation of tension, compression and bending zones as well as their combinations in the structures. Applying the traditional methods for determination of material characteristics in combination with linear deformation models (in particular those that do not account for the different strength and stiffness behavior in tension and compression) for calculation of such structures, one risks to distort the deformation and stress pattern significantly, which, in its turn, results in either underestimation or overestimation of the structure's strength and rigidity. Keeping in mind that carbon fibers are used for manufacturing of high-duty structures, their computation demands different strength and stiffness behavior in tension and compression to be taken into account.
