**4. Tsai-Hill failure criteria assessment of longitudinal tensile strength**

Failure theory is essential in determining whether the composite has failed. Literature review has shown that results of failure prediction depend on failure criterion applied and one major failure criteria used in the industries is Tsai-Hill and

direction of applied load [55]. Although Jones [56] intuitively suggested that highest value of material properties may not necessarily occur along the principal material directions, rather it is essential that transverse reinforcement is needed in unidirec-

As can be seen in **Figure 5** that shear modulus peaked at 45° fiber orientation and shear modulus was symmetric at about 45° fiber orientation angle for both PEFBFC and PPSFC considered. This implies that the higher in-plane shear resistance is achievable when fiber orientation is 45°. Also the respective minimum value of 3622.99 and 3332.83 MPa at fiber orientation 0° for PEFBFC and PPSFC can be seen to gradually increase to maximum values of 3843.57 and 3731.758 MPa at fiber orientation 45° and then reversed parabolically at 90° where it again reaches to 3622.99 and 3332.84 MPa. Similar trend was obtained by Farooq and Myler [58] who developed efficient procedures for determination of mechanical properties of carbon fiber-reinforced laminated composite panels. This trend in which the value of *Gxy* peaks at 45° fiber orientation angle and lowers at 0° and 90° fiber orientation angle indicates that off-axis reinforcement is very necessary for robust shear stiff-

**Figure 6** shows variation of Poisson's ratio with fiber orientation, the graph depicts a gradual drop of major Poisson's ratio (*vxy*Þ for PEFBFC and PPSFC respectively from 0.38 and 0.29 when fibers are aligned at 0° orientation angle to a lowest value of 0.18 and 0.15 value when fibers were aligned at 90° orientation angle. Additionally, the minor Poissons ratio (*vyx*Þ for PEFBFC and PPSFC increased respectively from 0.18 and 0.15 when fibers are aligned at 0° orientation angle to a highest value of 0.38 and 0.29 value when fibers were aligned at 90° orientation angle. **Figure 7** depicts the variation of shear coupling coefficient with fiber orientation, equal magnitude of shear coupling effect was obtained at 45° fiber orientation angle for both PEFBFC and PPSFC considered. Gibson [57] reported that shear coupling coefficient is a measure of the amount of shear strain developed in the *xy* plane per unit normal strain along the direction of the applied normal stress *σx*. **Figure 7** clearly indicate that the maximum value of the shear coupling coefficient in the reference *x*-direction for PEFBFC and PPSFC was attained at 30° fiber orientation angle while the coefficient in the reference *y*-direction for PEFBFC and

tional fiber composites which are subjected to multi axial loading [57].

*Composite and Nanocomposite Materials - From Knowledge to Industrial Applications*

ness in unidirectional composites [57].

**Figure 5.**

**192**

*Variation of shear modulus with fiber orientation in PEFBFC and PPSFC.*

failure criteria. Additionally, since composites ultimate tensile strength and strain depend on the fiber orientation, a failure criterion must be used in which the applied stress system is also in material axis [54]. Tsai-Hill theory considers an interaction of the stresses in the fiber direction. It postulates that failure can only occur in reinforced composites when the failure index exceeds 1, hence Eq. (19) must be satisfied to avoid failure.

By considering an arbitrary positive angle θ with reference to the *x*-axis in **Figure 3**, Ihueze et al. [43] transformed the stresses within the global axes (*x*-*y*) into material axes 1–2 as given in Eq. (15)

$$\begin{Bmatrix} \sigma\_1\\ \sigma\_2\\ \tau\_{12} \end{Bmatrix} = \begin{bmatrix} c^2 & s^2 & 2sc\\ s^2 & c^2 & -2sc\\ -sc & s c & (c^2 - s^2) \end{bmatrix} \begin{Bmatrix} \sigma\_{\chi}\\ \sigma\_{\mathcal{V}}\\ \tau\_{\chi\chi} \end{Bmatrix} \tag{15}$$

where *c* = *cosθ* and *s* = *sinθ*. Taking longitudinal direction stresses as *σ<sup>y</sup>* ¼ *τxy* ¼ 0 and thus

$$
\sigma\_1 = \sigma\_\text{x} \cos^2 \theta \tag{16}
$$

$$
\sigma\_2 = \sigma\_\text{x} \sin^2 \theta \tag{17}
$$

**5. Variation of tangential stress and modulus around a structural**

*Variation of longitudinal tensile strength with fiber orientation for PEFBFC and PPSFC.*

**S/N Orientation** *θ* **PEFBFC PPSFC** 0 410.15 288.1 5 195.86 152.635 10 108.786 86.8937 15 75.4556 60.6747 20 58.6356 47.3142 25 48.8262 39.5106 30 42.6439 34.6088 35 38.6038 31.4352 40 35.9607 29.4005 45 34.3043 28.1833 50 33.3926 27.5977 55 33.0716 27.5303 60 33.2311 27.9038 65 33.774 28.6492 70 34.5913 29.679 75 35.5418 30.8595 80 36.4447 31.9912 85 37.0995 32.8242 90 37.3397 33.133

*Strength Analysis and Variation of Elastic Properties in Plantain Fiber/Polyester Composites…*

*DOI: http://dx.doi.org/10.5772/intechopen.90890*

Structural discontinuity arising from holes in reinforced composites created for joining or access purposes causes stress concentration at the point of discontinuity [59].

**discontinuity**

**Figure 8.**

**195**

**Table 5.**

*Longitudinal tensile strength variation with fiber orientation angle.*

$$
\tau\_{12} = -\sigma\_{\text{x}} \cos \theta \sin \theta \tag{18}
$$

Considering Tsai-Hill failure criterion and setting the failure index as 1 for the composite failure to occur:

$$
\left(\frac{\sigma\_1}{S\_{u1}}\right)^2 + \left(\frac{\sigma\_2}{S\_{u2}}\right)^2 + \left(\frac{\tau\_{12}}{\tau\_{\max}}\right)^2 - \left(\frac{\sigma\_1}{S\_{u1}}\right)\left(\frac{\sigma\_2}{S\_{u1}}\right) = 1\tag{19}
$$

Substituting the appropriate value in Eq. (19) we have for PEFBFC

$$\left(\frac{\sigma\_{\text{x}}\cos^{2}\theta}{410.15}\right)^{2} + \left(\frac{\sigma\_{\text{x}}\sin^{2}\theta}{37.3397}\right)^{2} + \left(\frac{-\sigma\_{\text{x}}\cos\theta\sin\theta}{19.3100}\right)^{2} - \left(\frac{\sigma\_{\text{x}}\cos^{2}\theta}{410.15}\right)\left(\frac{\sigma\_{\text{x}}\sin^{2}\theta}{410.15}\right) = 1$$

$$\sigma\_{\text{x}, \text{PEFBFC}} = \sqrt{\frac{1}{\left(\cos^{4}\theta\_{\text{/410.15}^{2}} + \sin^{4}\theta\_{\text{/37.397}^{2}} + \cos^{2}\theta\sin^{2}\theta\_{\text{/3100}^{2}} - \cos^{2}\theta\sin^{2}\theta\_{\text{/410.15}^{2}}\right)}}$$

And for PPSFC

$$\left(\frac{\sigma\_x \cos^2 \theta}{288.10}\right)^2 + \left(\frac{\sigma\_x \sin^2 \theta}{33.1330}\right)^2 + \left(\frac{-\sigma\_x \cos \theta \sin \theta}{15.5700}\right)^2 - \left(\frac{\sigma\_x \cos^2 \theta}{288.10}\right)\left(\frac{\sigma\_x \sin^2 \theta}{288.10}\right) = 1$$

$$\sigma\_{\text{x, ppSFC}} = \sqrt{\frac{1}{\left(\cos^4 \theta \prime\_{288.10^2} + \sin^4 \theta \prime\_{31.330^2} + \cos^{20} \theta \sin^2 \theta \prime\_{15.5700^2} - \cos^{20} \theta \sin^2 \theta \prime\_{288.10^2}\right)}}$$

Hence the value for *σ<sup>x</sup>* is then calculated for orientation ranging from 0° to 90° as shown in **Table 5**.

The variation of longitudinal tensile strength with fiber orientation for PEFBFC and PPSFC has been presented in **Figure 8**, it can be seen that the tensile strength equals 410.15 and 288.1 MPa which are the longitudinal tensile strength for PEFBFC and PPSFC respectively when fiber orientation angle is 0°; on the other hand, the tensile strength equals 37.3397 and 33.133 MPa which are the transverse tensile strength for PEFBFC and PPSFC respectively when fiber orientation angle is 90°.


*Strength Analysis and Variation of Elastic Properties in Plantain Fiber/Polyester Composites… DOI: http://dx.doi.org/10.5772/intechopen.90890*

#### **Table 5.**

failure criteria. Additionally, since composites ultimate tensile strength and strain depend on the fiber orientation, a failure criterion must be used in which the applied stress system is also in material axis [54]. Tsai-Hill theory considers an interaction of the stresses in the fiber direction. It postulates that failure can only occur in reinforced composites when the failure index exceeds 1, hence Eq. (19)

*Composite and Nanocomposite Materials - From Knowledge to Industrial Applications*

By considering an arbitrary positive angle θ with reference to the *x*-axis in **Figure 3**, Ihueze et al. [43] transformed the stresses within the global axes (*x*-*y*) into

<sup>2</sup> 2*sc*

<sup>2</sup> ð Þ

� *<sup>σ</sup>*<sup>1</sup> *Su*<sup>1</sup> � � *σ*<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup>

� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup>

� �

<sup>33</sup>*:*1330<sup>2</sup> þ *cos* <sup>2</sup>*<sup>θ</sup> sin* <sup>2</sup>*<sup>θ</sup>*

Hence the value for *σ<sup>x</sup>* is then calculated for orientation ranging from 0° to 90°

<sup>37</sup>*:*3397<sup>2</sup> þ *cos* <sup>2</sup>*<sup>θ</sup> sin* <sup>2</sup>*<sup>θ</sup>*

3 7 5

8 ><

>:

*τ*<sup>12</sup> ¼ �*σxcosθsinθ* (18)

*Su*<sup>1</sup> � �

� � *σ<sup>x</sup> sin* <sup>2</sup>

<sup>19</sup>*:*3100<sup>2</sup> � *cos* <sup>2</sup>*<sup>θ</sup> sin* <sup>2</sup>*<sup>θ</sup>*

� � *σ<sup>x</sup> sin* <sup>2</sup>

<sup>15</sup>*:*5700<sup>2</sup> � *cos* <sup>2</sup>*<sup>θ</sup> sin* <sup>2</sup>*<sup>θ</sup>*

� *<sup>σ</sup><sup>x</sup> cos* <sup>2</sup>*<sup>θ</sup>* 410*:*15

*=*

� *<sup>σ</sup><sup>x</sup> cos* <sup>2</sup>*<sup>θ</sup>* 288*:*10

*=*

*σx σy τxy* 9 >=

>;

*θ* (16)

*θ* (17)

¼ 1 (19)

*θ* 410*:*15 � �

> *=*410*:*15<sup>2</sup>

*θ* 288*:*10 � �

> *=*288*:*102

¼ 1

¼ 1

(15)

<sup>2</sup> *<sup>c</sup>*<sup>2</sup> �2*sc* �*sc sc c*<sup>2</sup> � *<sup>s</sup>*

where *c* = *cosθ* and *s* = *sinθ*. Taking longitudinal direction stresses as *σ<sup>y</sup>* ¼ *τxy* ¼ 0

*<sup>σ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>σ</sup><sup>x</sup> cos* <sup>2</sup>

*<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>σ</sup><sup>x</sup> sin* <sup>2</sup>

Considering Tsai-Hill failure criterion and setting the failure index as 1 for the

<sup>þ</sup> *<sup>τ</sup>*<sup>12</sup> *τmax* � �<sup>2</sup>

<sup>þ</sup> �*σxcosθsin<sup>θ</sup>* 19*:*3100 � �<sup>2</sup>

<sup>þ</sup> �*σxcosθsin<sup>θ</sup>* 15*:*5700 � �<sup>2</sup>

*=*

The variation of longitudinal tensile strength with fiber orientation for PEFBFC and PPSFC has been presented in **Figure 8**, it can be seen that the tensile strength equals 410.15 and 288.1 MPa which are the longitudinal tensile strength for PEFBFC and PPSFC respectively when fiber orientation angle is 0°; on the other hand, the tensile strength equals 37.3397 and 33.133 MPa which are the transverse tensile strength for PEFBFC and PPSFC respectively when fiber orientation

Substituting the appropriate value in Eq. (19) we have for PEFBFC

*=*

*c*<sup>2</sup> *s*

*s*

2 6 4

must be satisfied to avoid failure.

and thus

composite failure to occur:

*σ<sup>x</sup> cos* <sup>2</sup>*θ* 410*:*15 � �<sup>2</sup>

*σ<sup>x</sup>*, PEFBFC ¼

And for PPSFC

*σ<sup>x</sup> cos* <sup>2</sup>*θ* 288*:*10 � �<sup>2</sup>

*σ<sup>x</sup>*, PPSFC ¼

as shown in **Table 5**.

angle is 90°.

**194**

*σ*1 *Su*<sup>1</sup> � �<sup>2</sup>

<sup>þ</sup> *<sup>σ</sup><sup>x</sup> sin* <sup>2</sup>

*cos*4θ*=*

<sup>þ</sup> *<sup>σ</sup><sup>x</sup> sin* <sup>2</sup>

*cos* <sup>4</sup> *θ=*

s

s

<sup>þ</sup> *<sup>σ</sup>*<sup>2</sup> *Su*<sup>2</sup> � �<sup>2</sup>

*θ* 37*:*3397 � �<sup>2</sup>

*θ* 33*:*1330 � �<sup>2</sup>

<sup>410</sup>*:*15<sup>2</sup> þ *sin* <sup>4</sup>*<sup>θ</sup>*

<sup>288</sup>*:*10<sup>2</sup> þ *sin* <sup>4</sup>*<sup>θ</sup>*

material axes 1–2 as given in Eq. (15)

*σ*1 *σ*2 *τ*12 9 >= >; ¼

8 ><

>:

*Longitudinal tensile strength variation with fiber orientation angle.*

**Figure 8.** *Variation of longitudinal tensile strength with fiber orientation for PEFBFC and PPSFC.*
