However, for the bending stress in the X-Y plane under off-axis loading, the result is significantly different: the ratios of the axial direct stress and the bending stress are 46.44% and 53.56%, respectively.
As shown in Table 3, in the X-Y plane, the axial direct stress possesses 100% of the superimposed stress in the on-axis loading case; however, under an off-axis load, the bending stress and the axial direct stress play nearly equivalent roles in the maximum superimposed stress.
In addition, in the X-Z plane, it shows that the superimposed stress in the off-axis loading case is equal to that in the on-axis loading case; in contrast, in the X-Y plane, the superimposed stress in the off-axis loading case is approximately 1.944 times that in the on-axis loading case.
For the HFRP lower chord members, the off-axis load is the critical loading condition controlling the X-Y plane stress.
In addition, the X-Y plane superimposed stress in the off-axis loading case is approximately 1.944 times that in the on-axis loading case.
It is noted that when H = 0, which corresponds to a section taken through the surface at the GHA origin on the vertical plane GA, Equation (5) reduces to A = a[G.sup.2] + bG + c and Equation (6) reduces to B = 0, indicating that the parabolic trace originally plotted in the x-y plane in Figure 1, curve (iii), is recovered unchanged (see Figure 4).
These points are the two roots of the quadratic equation and, by virtue of symmetry, are always located an equal distance either side of the GA (or GB) plane, that is, behind and in front of the original x-y plane! This accounts for the conjugate pairing of the roots--it now remains to find an analytical expression for their exact location in the complex (Argand) HG plane.
into and out of the original x-y plane along the normal though the point x = -[b/2a] i.e., along a line parallel to the Im(x) ([equivalent to] H) axis.
It is not widely appreciated that the simple parabola plotted in the x-y plane is actually a 'slice' of a more general three-dimensional hyperbolic paraboloid surface when complex solutions are accounted for.