PSI - Issue 78

Mauro Mazzei et al. / Procedia Structural Integrity 78 (2026) 1649–1656

1651

2.5. Elliptical paraboloid

2 2 + 2 2 − 2 = 0 2 2 − 2 2 − 2 = 0 2 2 + 2 2 −  2 2 =0 2 2 + 2 2 − 2 2 =0

2.6. Hyperbolic paraboloid

2.7. Generally express form where  = 1, -1, or 0.

2.8. Equation form 1

Comparing the equation [2.8] in which the denominator of y 2 is also a 2 , and not b 2 as in the previous [2.7]. Similarly, it is observed that [2.7] is obtained from surface [2.8] by uniform expansion with respect to the Oxz plane, with coefficient b/a (see Fig1).

Fig. 1. Oxz plane.

To generate the surface [2.8] we must generate any plane z = h, perpendicular to the Oz axis, and examine its intersection with the surface [2.8]. By placing z = h in equation [2.8] we obtain the equation:

2.9. Equation form 2

2 + 2 = 2 (  + ℎ 2 2 ) If  + h2/c2 is positive, the equation obtained, together with z = h, gives a circle lying in the plane z = h and with its centre on the Oz axis. If  + h2/c2 is negative, which can only be the case if  = -1 and h is sufficiently small, the plane z = h does not intrude the surface [2.8], since the sum of the squares x2 + Y2 cannot be a negative number. Therefore the surface [2.8] consists of circumferences lying on planes perpendicular to the Oz axis and having their centres on the Oz axis. But in this case, surface [2.8] is a surface of rotation about the Oz axis. Intersecting it with