Question 7.2: Repeat Example Problem 7.1, except use the Tsai–Wu criterion...

We once again perform the analysis described by the flow diagram shown in Figure 7.2. The solution process is identical to that described in Example Problem 7.1, until we reach step 7. In this latter step we apply the Tsai–Wu failure criterion rather than the maximum stress criterion. For plane stress conditions, the Tsai–Wu criterion is given by Equation 5.62, repeated here for convenience:

X_1\sigma _{11}+X_2\sigma _{22}+X_{11}\sigma _{11}^2+X_{22}\sigma _{22}^2+X_{66}\tau _{12}^2+2X _{12}\sigma _{11}\sigma _{22}\lt 1  (5.62)(repeated)

Using Equations 5.56, 5.57, 5.59, and 5.61, and the properties listed for graphite–epoxy in Table 3.1, we find

X_1 =\frac{1}{\sigma _{11}^{fT}} -\frac{1}{\sigma _{11}^{fC}}                  X_{11}=\frac{1}{\sigma _{11}^{fT}\sigma _{11}^{fC}}                       (5.56)

X_2 =\frac{1}{\sigma _{22}^{fT}} -\frac{1}{\sigma _{22}^{fC}}              X_{22}=\frac{1}{\sigma _{22}^{fT}\sigma _{22}^{fC}}                      (5.57)

X_{44}=\left\lgroup\frac{1}{\tau _{23}^f}\right\rgroup ^2 \quad X_{55}=\left\lgroup\frac{1}{\tau _{13}^f}\right\rgroup ^2 \quad X_{66}=\left\lgroup\frac{1}{\tau _{12}^f}\right\rgroup ^2                   (5.59)

X_{12}=\frac{-1}{2}\sqrt{X_{11}X_{22}}=\frac{-1}{2\sqrt{\sigma _{11}^{fT}\sigma _{11}^{fC}\sigma _{22}^{fT}\sigma _{22}^{fC}} }\\[0.5cm] X_{13}=\frac{-1}{2}\sqrt{X_{11}X_{33}}=\frac{-1}{2\sqrt{\sigma _{11}^{fT}\sigma _{11}^{fC}\sigma _{33}^{fT}\sigma _{33}^{fC}} }\\[0.5cm] X_{23}=\frac{-1}{2}\sqrt{X_{22}X_{33}}=\frac{-1}{2\sqrt{\sigma _{22}^{fT}\sigma _{22}^{fC}\sigma _{33}^{fT}\sigma _{33}^{fC}} }                                       (5.61)

We now apply the Tsai–Wu criterion to the 0°, 30°, and 60° plies in turn. First consider the 0° plies (i.e., plies 1 and 6). From Example Problem 7.1, the stresses induced in these plies are given by

Substituting these expressions and material constants into Equation 5.62, for the 0° plies the Tsai–Wu failure criterion becomes the following quadratic equation:

The two roots to this equation are

As the problem statement specified that the applied loading is tensile, we select the positive root (if the applied load was compressive we would select the negative root). Thus, according to the Tsai–Wu criterion for a failure to occur in the 0° plies the tensile unit load would have to be increased by a load factor of 345.1 × 10^3.

Repeating this process for the 30° and 60° plies results in critical unit load factors of 153.7 × 10^3 and 81.3 × 10^3, respectively. Hence, according to the Tsai–Wu criterion the first-ply failure load is N_{xx }= 81.3 kN/m, which corresponds to failure of the 60° plies (plies 3 and 4). Equivalently, the effective first-ply failure stress is

Note that first ply failure is predicted to occur in the 60° plies by both the maximum stress and the Tsai–Wu failure criterions. However, the first-ply failure loads are quite different: according to the maximum stress criterion, first ply failure will occur at a load level of N_{xx} = 123.0 kN/m, whereas according to the Tsai–Wu criterion first ply failure will occur at a load level of N_{xx} = 81.3 kN/m, a difference of over 50%.