**4.1 Fundamental stress analysis concepts**

All system designers must, at early stages, identify the probable failure modes, select a suitable parameter by which severity of loading and environment may be analytically represented, propose a material and geometry for components and implement critical strength properties related to the probable failure mode. The magnitude of the loading severity parameter must be calculated under applicable loading and environmental conditions, and compared with the critical strength property. Failure may be averted by assuring that the loading severity parameter is safely less than the corresponding critical strength property for each potential failure mode.

The most important loading severity parameters are stress, strain, and strain energy per unit volume. Of these, stress is usually selected for calculation purposes. To completely define the state of stress at any selected point within a solid body, it is necessary to describe the magnitudes and directions of stress vectors on all possible planes that could be passed through such point. One way of defining the state of stress at a point is to determine all stress components that can occur on the faces of an infinitesimal cube of material placed at the origin of an arbitrarily selected right-handed Cartesian coordinate system of known orientation. Each of these components of stress may be classified as either a normal stress *σ* normal to a face of the cube, or a shear stress *τ* parallel to a face of the cube. The illustration in **Figure 8** depicts all possible stress components acting on an infinitesimal cubic volume element of dimensions *dx-dy-dz*.

Depending upon whether its material behaves in a *brittle* or a *ductile* manner, failure at the governing critical point of a component is dependent upon the principal normal stresses, the principal shearing stresses, or some combination of these. In any event, the designer must evaluate principal normal stresses and principal shearing stresses for any combination of applied loads. To do this, the general stress cubic Eq. (1) may be employed to find the principal stresses *σ*1, *σ*2, *σ*<sup>3</sup> and as a function of the readily calculable components of stress *σx*, *σy*, *σz*, *τxy*, *τyz*, *τzx* relative to any selected *x-y-z* coordinate system. The general stress cubic equation, developed from equilibrium concepts, has the form – see [17]:

$$\begin{aligned} \sigma^3 - \sigma^2 \left( \sigma\_{\rm x} + \sigma\_{\rm y} + \sigma\_{\rm z} \right) + \sigma \left( \sigma\_{\rm x} \sigma\_{\rm y} + \sigma\_{\rm y} \sigma\_{\rm x} + \sigma\_{\rm x} \sigma\_{\rm x} - \tau\_{\rm xy}^2 - \tau\_{\rm yx}^2 - \tau\_{\rm xx}^2 \right) - \\ - \left( \sigma\_{\rm x} \sigma\_{\rm y} \sigma\_{\rm x} + 2 \sigma\_{\rm x} \sigma\_{\rm y} \tau\_{\rm xx} - \sigma\_{\rm x} \tau\_{\rm yx}^2 - \sigma\_{\rm y} \tau\_{\rm xx}^2 - \sigma\_{\rm x} \tau\_{\rm xy}^2 \right) = 0 \end{aligned} \tag{1}$$

Since all normal and shearing stress components are real numbers, all three roots of the general stress cubic equation are real. These three roots are the principal normal stresses *σ*1, *σ*2, *σ*3. It is also possible to find the directions of principal stress vectors and principal shearing stress vectors if necessary. Furthermore, it can be shown that the magnitudes of the principal shearing stresses may be calculated from Eq. (2):

**Figure 8.** *Complete definition of the state of stress at arbitrary points.*

$$|\tau\_1| = \left|\frac{\sigma\_2 - \sigma\_3}{2}\right|, \quad |\tau\_2| = \left|\frac{\sigma\_3 - \sigma\_1}{2}\right|, \quad |\tau\_3| = \left|\frac{\sigma\_1 - \sigma\_2}{2}\right|\tag{2}$$

To summarize, if loads and geometry are known for a component in any system, the designer may identify the critical point, arbitrarily select a convenient *x-y-z* coordinate system, and calculate the resultant six stress components *σx*, *σy*, *σz*, *τxy*, *τyz*, *τzx*. The above equations can be solved to find the principal normal stresses and the principal shearing stresses.

#### **4.2 Failure criteria for composite materials**

To determine whether a laminate will fail due to any applied load, the stresses across the different plies needs to be calculated and next a failure criterion based on these stress levels must be selected. The failure of composites occurs in multiple steps. When the stress in the first ply or first group of plies is high enough, it fails. This point of failure is the first ply failure, beyond which a laminate can still carry the load. For a safe design, laminates should be exposed to stresses below this point. The point where the total failure occurs is termed the ultimate laminate failure. Failure of composites occurs on a micromechanical scale due to fiber damage, matrix cracking, or interface or inter-phase failure. The local failure modes mentioned above cannot predict global laminate failure satisfactorily. Composite failure theories predict global laminate failure. These failure theories can be interactive, non-interactive or partially interactive. The non-interactive theories do not consider the interaction between different stress components, whereas the interactive theories do. The three theories available for laminate failure criteria are:


The *Maximum Stress Criterion* widely applies to composite shells. Failure occurs according to the maximum stress criterion when the stress in one of the principal material directions exceeds the strength in that direction. The overall state of stress in the global coordinates is first computed by the program. Then, the program computes stress along the principal material directions for each lamina by applying a coordinate transformation. The program assumes a state of plane stress - 2D - for a lamina with *σ*<sup>3</sup> ¼ 0, *τ*<sup>13</sup> ¼ 0, *τ*<sup>23</sup> ¼ 0. The failure index *FI* is computed as follows:

$$FI = \max\left(\frac{\sigma\_1}{\mathcal{S}\_1}, \frac{\sigma\_2}{\mathcal{S}\_2}, \left|\frac{\sigma\_{12}}{\mathcal{S}\_{12}}\right|\right) \tag{3}$$

The software reports a Factor of Safety *FOS* <sup>¼</sup> <sup>1</sup> *FI* <sup>¼</sup> <sup>1</sup> max *<sup>σ</sup>*<sup>1</sup> *S*1 , *σ*2 *S*2 , *<sup>σ</sup>*<sup>12</sup> *S*12 . The

composite will not fail if *FOS*>1. The significance of the parameters involved in the equations above is:


There are some additional requirements:


The maximum stress criterion has the following characteristics:


The *Tsai-Hill Criterion* considers the distortion energy portion of the total strain energy that is stored due to loading. The distortion energy is the portion of strain energy that causes shape change. The other portion is the dilatation energy that causes volume or area change due to loading. For composite shells each lamina is assumed to be in a state of plane stress with *σ*<sup>3</sup> ¼ 0, *τ*<sup>13</sup> ¼ 0, *τ*<sup>23</sup> ¼ 0. The failure index is computed as:

$$FI = \frac{\sigma\_1^2}{S\_1^2} - \frac{\sigma\_1 \sigma\_2}{S\_1^2} + \frac{\sigma\_2^2}{S\_2^2} + \frac{\tau\_{12}^2}{S\_{12}^2} \tag{4}$$

The significance of the parameters involved in Eq. (4) above is:


The program reports a Factor of Safety *FOS* <sup>¼</sup> <sup>1</sup>ffiffiffi *FI* <sup>p</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *σ*2 1 *S*2 1 �*σ*1*σ*<sup>2</sup> *S*2 1 þ *σ*2 2 *S*2 2 þ*τ*2 12 *S*2 <sup>12</sup> <sup>r</sup> .

The *FOS* is the coefficient by which all stress components should be multiplied to reach laminate failure at *FI* ¼ 1. The composite will not fail if *FOS*>1. There are also some additional requirements for this failure criterion:


The Tsai-Hill criterion considers the interaction between different stress components. However, it cannot predict various failure modes including fiber failure, matrix failure, and fiber-matrix interface failure.

The *Tsai-Wu Criterion* is applied to determine the factor of safety for composite orthotropic shells. This criterion considers the total strain energy - both distortion energy and dilatation energy - for predicting failure. It is more general than the Tsai-Hill failure criterion because it distinguishes between compressive and tensile failure strengths. For a 2D state plane stress - *σ*<sup>3</sup> ¼ 0, *τ*<sup>13</sup> ¼ 0, *τ*<sup>23</sup> ¼ 0 - the Tsai-Wu failure criterion is expressed as:

$$F\_1\sigma\_1 + F\_2\sigma\_2 + 2F\_{12}\sigma\_1\sigma\_2 + F\_{11}\sigma\_1^2 + F\_{22}\sigma\_2^2 + F\_6\tau\_{12} + F\_{66}\tau\_{12}^2 = 1\tag{5}$$

The coefficients *Fij* of the orthotropic Tsai-Wu failure criterion are related to the material strength parameters of the lamina determined by experiments. They are calculated from with following equations:

$$F\_{1} = \left(\frac{1}{S\_{1\text{ }Tensile}} - \frac{1}{S\_{1\text{ }Compressive}}\right), F\_{2} = \left(\frac{1}{S\_{1\text{ }Tensile}^{2}} - \frac{1}{S\_{1\text{ }Compressive}^{2}}\right),$$

$$F\_{12} = -\frac{1}{2}\sqrt{\frac{1}{S\_{1\text{ }Tensile}^{2} \cdot S\_{1\text{ }Compressive}^{2}} \cdot \frac{1}{S\_{2\text{ }Tensile}^{2} \cdot S\_{2\text{ }Compressive}^{2}}},$$

$$F\_{11} = \frac{1}{S\_{1\text{ }Compressive}^{2} \cdot S\_{1\text{ }Tensile}^{2}}, F\_{22} = \frac{1}{S\_{2\text{ }Compressive}^{2} \cdot S\_{2\text{ }Tensile}^{2}},$$

$$F\_{6} = \left(\frac{1}{S\_{12\text{ }T}^{2}} - \frac{1}{S\_{12\text{ }C}^{2}}\right), F\_{66} = \frac{1}{S\_{12\text{ }C} \cdot S\_{12\text{ }T}}\tag{6}$$

The significance of the parameters involved in the equations above is:


The factor of safety *FOS* is the coefficient by which all laminate stress components should be multiplied to reach laminate failure according to the Tsai-Wu criterion stated above. The *FOS* for laminate failure is calculated as:

$$\begin{aligned} FOS &= \frac{R - C\_1}{2C\_2}, C\_1 = F\_1\sigma\_1 + F\_2\sigma\_2 + F\_6\tau\_{12}, \\\\ C\_2 &= F\_{11}\sigma\_1^2 + F\_{22}\sigma\_2^2 + F\_{66}\tau\_{12}^2 + 2F\_{12}\sigma\_1\sigma\_2, \\\ \end{aligned} \tag{7}$$

If *FOS*>1, the composite will not fail. The Tsai-Wu failure criterion cannot predict different failure modes including fiber failure, matrix failure, and fibermatrix interface failure.

### **4.3 Example**

Consider a composite material with natural fibers. The composite has seven plies, a layup [0/90/0/90/0/90/0] and it is subject to three point bending with support span of 20 mm by applying a uniform load *P* = 10 N/mm at the center. The objective of the simulation is to evaluate the longitudinal stress SX, the displacement UY at point B and the transverse shear stress TXY at point C. The geometric and mechanical properties of the composite as evaluated by measurement and experiments are:


**Figure 9.** *Plies orientation.*


The simulation results are shown in **Figure 10** and are validated by experiments.

**Figure 10.** *Stress, displacement and strain diagrams.*
