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

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

**Figure 9.** *Depiction of hole in the plantain fiber-reinforced composites sample.*

Adequate comprehension of stress redistribution pattern and concentrations is helpful for proficient and safe structural designs. Unlike in ductile materials where stress concentration is of no much ado, plantain fiber-reinforced composites may be sufficiently brittle, hence every form of stress concentration and structural discontinuity has to be properly designed. In a typical scenario where a circular hole is created in the composite as shown in **Figure 9**, assuming no interlaminar stresses exist around the free edge of the hole, the ply is nominally stressed by *σ*1, *σ*2, *σ*<sup>12</sup> some distance away from the hole as indicated. Lekhnitskii [60] derived various useful expressions for stress distribution around holes in a composite plate, the tangential elastic modulus *E*<sup>∅</sup> at an angular position ∅ is determined using Eq. (20).

$$E\_{\mathcal{Q}} = \frac{1}{\left(\frac{\sin^4 \mathcal{Q}}{E\_1} + \left[\frac{1}{G\_{12}} - \frac{2w\_1}{E\_1}\right] \sin^2 \mathcal{Q} \cos^2 \mathcal{Q} + \frac{\cos^4 \mathcal{Q}}{E\_2}\right)}\tag{20}$$

Hence the tangential stress *σ*<sup>∅</sup> at the perifery of the hole with an angle ∅ is found from Eq. (21).

$$
\sigma\_{\mathcal{Q}} = \frac{E\_{\mathcal{Q}}}{E\_1} (A\sigma\_1 + B\sigma\_2 + C\sigma\_{12}) \tag{21}
$$

At the edge of the hole, only the tangential stress *σ*<sup>∅</sup> > 0, thus *σ<sup>r</sup>* ¼ *σr*<sup>∅</sup> ¼ 0 in

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

*<sup>σ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>σ</sup><sup>x</sup>* <sup>¼</sup> *<sup>σ</sup>*<sup>∅</sup> *sin* <sup>2</sup>

*<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>σ</sup><sup>y</sup>* <sup>¼</sup> *<sup>σ</sup>*<sup>∅</sup> *cos* <sup>2</sup>

*σ*1 *Su*<sup>1</sup> 

*σ*2 *Su*<sup>2</sup> 

*σ*<sup>12</sup> *τmax* 

 

 

> 

**S/N Eø σ<sup>ø</sup> σ<sup>1</sup> σ<sup>2</sup> σ<sup>12</sup> F.I.1 F.I.2 F.I.12 Stress conc.** 0 7030.962 �23.338 0.000 �23.338 0.000 0.000 0.625 0.000 �0.69 5 7053.373 �22.807 �0.173 �22.634 1.980 0.000 0.606 0.103 �0.67 10 7121.323 �21.211 �0.640 �20.571 3.627 0.002 0.551 0.188 �0.62 15 7236.950 �18.542 �1.242 �17.300 4.636 0.003 0.463 0.240 �0.55 20 7403.786 �14.786 �1.730 �13.056 4.752 0.004 0.350 0.246 �0.43 25 7626.692 �9.917 �1.771 �8.146 3.799 0.004 0.218 0.197 �0.29 30 7911.712 �3.903 �0.976 �2.927 1.690 0.002 0.078 0.088 �0.11 35 8265.787 3.303 1.087 2.217 �1.552 0.003 0.059 0.080 0.10 40 8696.213 11.751 4.855 6.896 �5.786 0.012 0.185 0.300 0.35 45 9209.655 21.484 10.742 10.742 �10.742 0.026 0.288 0.556 0.63 50 9810.462 32.515 19.080 13.434 �16.010 0.047 0.360 0.829 0.96 55 10497.994 44.784 30.050 14.733 �21.042 0.073 0.395 1.090 1.32 60 11262.726 58.103 43.577 14.526 �25.159 0.106 0.389 1.303 1.71 65 12081.304 72.076 59.203 12.873 �27.607 0.144 0.345 1.430 2.12 70 12911.622 86.025 75.962 10.063 �27.648 0.185 0.269 1.432 2.53 75 13690.476 98.951 92.323 6.628 �24.738 0.225 0.178 1.281 2.91 80 14337.671 109.599 106.294 3.305 �18.742 0.259 0.089 0.971 3.22 85 14769.765 116.668 115.782 0.886 �10.130 0.282 0.024 0.525 3.43 90 14922.000 119.152 119.152 0.000 0.000 0.291 0.000 0.000 3.50 95 14769.765 116.668 115.782 0.886 10.130 0.282 0.024 0.525 3.43 100 14337.671 109.599 106.294 3.305 18.742 0.259 0.089 0.971 3.22 105 13690.476 98.951 92.323 6.628 24.738 0.225 0.178 1.281 2.91

*Variation of tangential stress, material axis stress and tangential modulus at the edge of material discontinuity*

in the material axes exceeds their respective ultimate strenght. Such that

Using the maximum stress criterion, the material will fail when any stress value

∅ (23)

∅ (24)

<1 (26)

<1 (27)

<1 (28)

*σ*<sup>12</sup> ¼ *σxy* ¼ �*σ*<sup>∅</sup> *cos* ∅ *sin* ∅ (25)

Eq. (22), therefore

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

**Table 6.**

**197**

*in PEFBFC.*

where

$$A = \cos^2 \mathcal{Q} + (1+p)\sin^2 \mathcal{Q}$$

$$B = q\left\{(q+p)\cos^2 \mathcal{Q} - \sin^2 \mathcal{Q}\right\}$$

$$C = (1+q+p)p\sin 2\mathcal{Q}$$

$$p = \sqrt{2(q-v\_{12}) + \frac{E\_1}{G\_{12}}}$$

$$q = \sqrt{\frac{E\_1}{E\_2}}$$

Using the stress transformation matrix and replacing the axes system 1–2 by radial (*r*)-tangential (∅), we can resolve the tangential stress *σ*<sup>∅</sup> back into the material axesin Eq. (22) for proper strength evaluation

$$\begin{Bmatrix} \sigma\_r \\ \sigma\_\oslash \\ \tau\_{r\oslash} \end{Bmatrix} = \begin{bmatrix} c^2 & s^2 & -2sc \\ s^2 & c^2 & 2sc \\ sc & -sc & (c^2 - s^2) \end{bmatrix} \begin{Bmatrix} \sigma\_\times \\ \sigma\_\nearrow \\ \tau\_{\ge\circ} \end{Bmatrix} \tag{22}$$

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

At the edge of the hole, only the tangential stress *σ*<sup>∅</sup> > 0, thus *σ<sup>r</sup>* ¼ *σr*<sup>∅</sup> ¼ 0 in Eq. (22), therefore

$$
\sigma\_1 = \sigma\_\ge = \sigma\_\mathcal{Q} \sin^2 \mathcal{Q} \tag{23}
$$

$$
\sigma\_2 = \sigma\_\chi = \sigma\_\mathcal{\otimes} \cos^2 \mathcal{\otimes} \tag{24}
$$

$$
\sigma\_{12} = \sigma\_{\text{xy}} = -\sigma\_{\mathcal{Q}} \cos \mathcal{Q} \sin \mathcal{Q} \tag{25}
$$

Using the maximum stress criterion, the material will fail when any stress value in the material axes exceeds their respective ultimate strenght. Such that

$$\left|\frac{\sigma\_1}{\mathbf{S}\_{u1}}\right| < \mathbf{1} \tag{26}$$

$$\left|\frac{\sigma\_2}{\mathbf{S}\_{u2}}\right| < 1\tag{27}$$

$$\left|\frac{\sigma\_{12}}{\sigma\_{\max}}\right| < 1\tag{28}$$


#### **Table 6.**

Adequate comprehension of stress redistribution pattern and concentrations is helpful for proficient and safe structural designs. Unlike in ductile materials where stress concentration is of no much ado, plantain fiber-reinforced composites may be sufficiently brittle, hence every form of stress concentration and structural discontinuity has to be properly designed. In a typical scenario where a circular hole is created in the composite as shown in **Figure 9**, assuming no interlaminar stresses exist around the free edge of the hole, the ply is nominally stressed by *σ*1, *σ*2, *σ*<sup>12</sup> some distance away from the hole as indicated. Lekhnitskii [60] derived various useful expressions for stress distribution around holes in a composite plate, the tangential elastic modulus *E*<sup>∅</sup>

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

at an angular position ∅ is determined using Eq. (20).

*Depiction of hole in the plantain fiber-reinforced composites sample.*

from Eq. (21).

**Figure 9.**

where

**196**

*<sup>E</sup>*<sup>∅</sup> <sup>¼</sup> <sup>1</sup> *sin* <sup>4</sup>∅ *<sup>E</sup>*<sup>1</sup> <sup>þ</sup> <sup>1</sup>

> *<sup>σ</sup>*<sup>∅</sup> <sup>¼</sup> *<sup>E</sup>*<sup>∅</sup> *E*1

*<sup>A</sup>* <sup>¼</sup> *cos* <sup>2</sup>

*p* ¼

material axesin Eq. (22) for proper strength evaluation

9 >= >; ¼

*σr σ*∅ *τ<sup>r</sup>*<sup>∅</sup>

8 ><

>:

*<sup>B</sup>* <sup>¼</sup> *q q*ð Þ <sup>þ</sup> *<sup>p</sup> cos* <sup>2</sup>

r

*q* ¼

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

*s*

2 6 4

Using the stress transformation matrix and replacing the axes system 1–2 by radial (*r*)-tangential (∅), we can resolve the tangential stress *σ*<sup>∅</sup> back into the

*<sup>G</sup>*<sup>12</sup> � <sup>2</sup>*v*<sup>12</sup> *E*1 h i

*sin* <sup>2</sup>

Hence the tangential stress *σ*<sup>∅</sup> at the perifery of the hole with an angle ∅ is found

<sup>∅</sup> <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>p</sup> sin* <sup>2</sup>

∅ � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð Þþ *q* � *v*<sup>12</sup>

> ffiffiffiffiffi *E*1 *E*2 r

<sup>2</sup> �2*sc*

<sup>2</sup> ð Þ

3 7 5

8 ><

>:

*σx σy τxy* 9 >=

>;

(22)

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

*C* ¼ ð Þ 1 þ *q* þ *p p sin* 2∅

<sup>∅</sup> � *sin* <sup>2</sup>

*E*1 *G*<sup>12</sup>

<sup>∅</sup> *cos* <sup>2</sup><sup>∅</sup> <sup>þ</sup> *cos* <sup>4</sup><sup>∅</sup>

ð Þ *Aσ*<sup>1</sup> þ *Bσ*<sup>2</sup> þ *Cσ*<sup>12</sup> (21)

∅

� � (20)

*E*2

*Variation of tangential stress, material axis stress and tangential modulus at the edge of material discontinuity in PEFBFC.*


The left hand side of Eqs. (26)–(28) represents the failure indices (FI). The maximum failure index (FI) for the applied stress is factored in to obtain the load factor. Due to the inherent material orthotropy, the failure zone of the plantain

fiber-reinforced composite as a result of structural discontinuity may not necessarily occur at the point of maximum stress concentration, therefore it is important to assess the extent of variation of the tangential stress around the hole edge and the failure index using maximum stress theory at other points aside the point of maximum stress concentration. Also in the present consideration we take a simplified scenario where *σ*<sup>2</sup> ¼ *σ*<sup>12</sup> ¼ 0 such that the ply of dimensions 150 � 19.05 � 3.2 mm with a circular hole at the center is subjected to only nominal axial stress *σ*PEFBFC ¼ 34 MPa and *σ*PPSFC ¼ 29 MPa. **Tables 6** and **7** depict the values of tangential stress, material axis stress and tangential modulus as computed using Eq. (20)–(25).

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

Tangential stress distribution at hole edge for PEFBFC and PPSFC are shown in **Figure 10**, the maximum stress value of 119.15 and 100.587 MPa was attained at angular position ø = 90° for PEFBFC and PPSFC respectively. However, considering various failure indices in **Tables 5** and **6**, failure will be initiated at ø = 70° for PEFBFC with stress concentration factor of 2.53 and ø = 65° for PPSFC with stress concentration factor of 2.13 which are less than the stress concentration around the

The utilization of plantain fiber-reinforced composites in structural applications empowers architects to acquire huge accomplishments in the usefulness, security and economy of development. These materials have high proportion of strength-todensity ratio, can be tailored to posses certain mechanical properties. The elastic constants of plantain fiber-reinforced composites depend greatly on fiber orientation with notable anisotropic characteristics which makes it less attractive for applications involving lugs and fittings. The present report amplified some notable design procedures in handling such limitations in plantain fiber-reinforced composites using relevant failure theories. Both plantain EFBFRC and PSFRC showed similar trends in response to the design scenario considered. Be that as it may be, a proficient utilization of plantain fiber-reinforced composites in structural

\* and Christopher Chukwutoo Ihueze<sup>2</sup>

1 Department of Mechanical Engineering, Nnamdi Azikiwe University, Awka,

2 Department of Industrial, Production Engineering, Nnamdi Azikiwe University,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

applications requires a cautious assessment of all influential factors.

\*Address all correspondence to: ce.okafor@unizik.edu.ng

provided the original work is properly cited.

peak stress when angular position is 90°.

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

**6. Conclusions**

**Author details**

Nigeria

**199**

Awka, Nigeria

Christian Emeka Okafor<sup>1</sup>

#### **Table 7.**

*Variation of tangential stress, material axis stress and tangential modulus at the edge of material discontinuity in PPSFC.*

**Figure 10.** *Tangential stress distribution at a hole edge for PEFBFC and PPSFC.*

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

fiber-reinforced composite as a result of structural discontinuity may not necessarily occur at the point of maximum stress concentration, therefore it is important to assess the extent of variation of the tangential stress around the hole edge and the failure index using maximum stress theory at other points aside the point of maximum stress concentration. Also in the present consideration we take a simplified scenario where *σ*<sup>2</sup> ¼ *σ*<sup>12</sup> ¼ 0 such that the ply of dimensions 150 � 19.05 � 3.2 mm with a circular hole at the center is subjected to only nominal axial stress *σ*PEFBFC ¼ 34 MPa and *σ*PPSFC ¼ 29 MPa. **Tables 6** and **7** depict the values of tangential stress, material axis stress and tangential modulus as computed using Eq. (20)–(25).

Tangential stress distribution at hole edge for PEFBFC and PPSFC are shown in **Figure 10**, the maximum stress value of 119.15 and 100.587 MPa was attained at angular position ø = 90° for PEFBFC and PPSFC respectively. However, considering various failure indices in **Tables 5** and **6**, failure will be initiated at ø = 70° for PEFBFC with stress concentration factor of 2.53 and ø = 65° for PPSFC with stress concentration factor of 2.13 which are less than the stress concentration around the peak stress when angular position is 90°.

## **6. Conclusions**

The left hand side of Eqs. (26)–(28) represents the failure indices (FI). The maximum failure index (FI) for the applied stress is factored in to obtain the load factor. Due to the inherent material orthotropy, the failure zone of the plantain

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

**S/N Eø σ<sup>ø</sup> σ<sup>1</sup> σ<sup>2</sup> σ<sup>12</sup> F.I.1 F.I.2 F.I.12 Stress conc.** 0 6817.175 20.978 0.000 20.978 0.000 0.000 0.633 0.000 0.72 5 6830.651 20.459 0.155 20.304 1.776 0.001 0.613 0.114 0.71 10 6872.014 18.909 0.570 18.339 3.234 0.002 0.554 0.208 0.65 15 6944.024 16.346 1.095 15.251 4.086 0.004 0.460 0.262 0.56 20 7051.119 12.791 1.496 11.295 4.111 0.005 0.341 0.264 0.44 25 7199.198 8.269 1.477 6.792 3.167 0.005 0.205 0.203 0.29 30 7395.333 2.793 0.698 2.095 1.209 0.002 0.063 0.078 0.10 35 7647.394 3.635 1.196 2.439 1.708 0.004 0.074 0.110 0.13 40 7963.509 11.024 4.555 6.469 5.428 0.016 0.195 0.349 0.38 45 8351.207 19.391 9.695 9.695 9.695 0.034 0.293 0.623 0.67 50 8816.012 28.736 16.863 11.873 14.150 0.059 0.358 0.909 0.99 55 9359.192 39.014 26.179 12.835 18.331 0.091 0.387 1.177 1.35 60 9974.406 50.087 37.565 12.522 21.688 0.130 0.378 1.393 1.73 65 10643.308 61.651 50.640 11.011 23.614 0.176 0.332 1.517 2.13 70 11330.958 73.175 64.615 8.560 23.518 0.224 0.258 1.510 2.52 75 11983.277 83.856 78.239 5.617 20.964 0.272 0.170 1.346 2.89 80 12530.185 92.667 89.873 2.794 15.847 0.312 0.084 1.018 3.20 85 12897.622 98.526 97.778 0.748 8.554 0.339 0.023 0.549 3.40 90 13027.500 100.587 100.587 0.000 0.000 0.349 0.000 0.000 3.47

*Variation of tangential stress, material axis stress and tangential modulus at the edge of material discontinuity*

**Table 7.**

*in PPSFC.*

**Figure 10.**

**198**

*Tangential stress distribution at a hole edge for PEFBFC and PPSFC.*

The utilization of plantain fiber-reinforced composites in structural applications empowers architects to acquire huge accomplishments in the usefulness, security and economy of development. These materials have high proportion of strength-todensity ratio, can be tailored to posses certain mechanical properties. The elastic constants of plantain fiber-reinforced composites depend greatly on fiber orientation with notable anisotropic characteristics which makes it less attractive for applications involving lugs and fittings. The present report amplified some notable design procedures in handling such limitations in plantain fiber-reinforced composites using relevant failure theories. Both plantain EFBFRC and PSFRC showed similar trends in response to the design scenario considered. Be that as it may be, a proficient utilization of plantain fiber-reinforced composites in structural applications requires a cautious assessment of all influential factors.

### **Author details**

Christian Emeka Okafor<sup>1</sup> \* and Christopher Chukwutoo Ihueze<sup>2</sup>

1 Department of Mechanical Engineering, Nnamdi Azikiwe University, Awka, Nigeria

2 Department of Industrial, Production Engineering, Nnamdi Azikiwe University, Awka, Nigeria

\*Address all correspondence to: ce.okafor@unizik.edu.ng

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
