Assignment #8
Given:
Nx=P2πR=6.37Nm
Nxy=T2πR2=255NmNy=Mx=My=Mxy=0.
For the layer sequence of ±30/03s the principle stresses for all the layers are shown in Table 1.
Table 1: Principle stresses of all the layers
σ1 / σ2 / τ12+30° / 0.00291p+0.739t / -0.000122p-0.0442t / -0.000257p+0.0246t
-30° / 0.00291p-0.739t / -0.000122p+0.0442t / 0.000257p+0.0246t
0° / 0.00549p / -0.000276p / 0.0491t
Substituting the values of principle stresses as well as Fi and Fij from Table 10.1 of the text book into the Tsai-Wu Criterion equation which has the form of
F1σ1+F2σ2+F11σ12+F22σ22+F66τ122-F11F22σ1σ2=1
and rearranging the result in the form of
Ap2+Bt2+Cpt+Dp+Et+F=0
for each layer, the constant can be calculated, Table 2.
Table 2: Constants of Tsai-Wu Criterion equation for all the layers.
+30° / -30° / 0°A / 15.199×10-12 / 15.199×10-12 / 34.744×10-12
B / 0.785×10-6 / 0.785×10-6 / 0.241×10-6
C / 3.704×10-9 / -3.704×10-9 / 0.
D / -2.218×10-6 / -2.218×10-6 / -4.872×10-6
E / -761.509×10-6 / 761.509×10-6 / 0.
F / -1 / -1 / -1
Figure 1 shows the three resulting ellipses. The common area of all the ellipses is the failure envelope (safe load region).
Figure 1: Tsai-Wu ellipses for all the layers
To compare the result with Maximum Stress Criterion 18 lines identified by the inequalities on the Table 3 should be drawn and the common region be selected as the failure envelope. The shaded area on Figure 2 illustrates this region.
Table 3: Limits of the principle stresses in all the layers.
σ1 / σ2 / τ12+30° / -1250<0.00291p+0.739t<1500 / -200<-0.000122p-0.0442t<50 / -100<-0.000257p+0.0246t<100
-30° / -1250<0.00291p-0.739t<1500 / -200<-0.000122p+0.0442t<50 / -100<0.000257p+0.0246t<100
0° / -1250<0.00549p<1500 / -200<-0.000276p<50 / -100<0.0491t<100
Figure 2: The safe load region for all the layers based on Maximum Stress Criterion.
Putting the result of the two criteria on one diagram with the same scale shows that Tsai-Wu is more conservative, Figure 3.
Figure 3: Superposition of Tsai-Wu and Maximum Stress Criterion