Question 7.3: Predict the last-ply failure load using the ply discount sch...

As determined in Example Problem 7.1, the 60° plies (plies 3 and 4) are the first plies to fail, and the first-ply failure load is N_{xx }= 122.9 kN/m, or equivalently, the first-ply failure stress is \overline{\sigma }_{xx} = 164 MPa. The initial elastic modulus was also determined to be \overline{E} _{xx}^{ex} = 82 9. GPa, so at the firstply failure load the axial strain is ε_{xx} = 164 MPa/82.9 GPa = 1978 μm/m. As per the flow diagram shown in Figure 7.3, we now discount the properties of the failed 60° plies. That is, we assume that the stiffnesses of plies 3 and 4 are now:

We repeat steps 2 through 7. The new [ABD] matrix, formed using the original stiffnesses for the 0° and 30° plies but with reduced stiffnesses for the 60° plies, is found to be

The new effective extensional modulus is \overline{E}_{xx}^{ex} = 79.6 GPa, a modest decrease from the original value \overline{E}_{xx}^{ex} = 82.9 GPa.

The process then proceeds as before: the new thermal stress and moment resultants are calculated (step 3), thermal stresses and strains for each ply are determined (step 4), ply stresses and strains caused by the unit load are determined and summed with the thermal stresses/ strains (steps 5 and 6), and the failure criterion applied, in this case the maximum stress failure criterion (step 7). It is predicted that the 30° plies (plies 2 and 5) will fail at a load N_{xx }= 199.0 kN/m, or equivalently, at an effective stress\overline{\sigma }_{xx} = 265 MPa. These failures occur at an axial strain ε_{xx} = 265 MPa/79.6 GPa = 3329 μm m/m.

Finally, the stiffnesses properties of the 30° plies are reduced and the entire process repeated still again. It is found that the effective extensional modulus is reduced to E_{xx}^{ex} = 73 1. GPa, and that the 0° plies will fail at a last-ply failure load N_{xx} = 292.4 kN/m, corresponding to an effective stress \overline{\sigma }_{xx} = 390 MPa and axial strain ε_{xx} = 5334 μm/m.

As a uniaxial tensile load was considered in this problem, the stress– strain pairs predicted at each ply failure load can be plotted to produce a predicted stress–strain curve to failure for the [0/30/60]_s graphite–epoxy laminate, as shown in Figure 7.4.