w(x, t) is the transverse displacement, [theta](x, t) is the rotation of the cross section due to bending, f(x, t) is the external transverse force, [tau](x, t) is the external bending moment, E is Young's modulus, G is the shear modulus, [rho] is the mass density, k is the shear coefficient factor, A is the cross-sectional area, and I is the area moment of inertia. The natural and geometric boundary conditions relevant to (1) are given by

The geometric and material properties of the example beam are as follows: length L = 4.352 m, cross-sectional area A = 1.31 x [10.sup.-3] [m.sup.2], area moment of inertia I = 5.71 x [10.sup.-7] [m.sup.4], Young's modulus E = 2.02 x [10.sup.11] N/[m.sup.2], shear modulus G = 7.7 x [10.sup.10] N/[m.sup.2], mass density [rho] = 15267 kg/[m.sup.3], and shear correction factor k = 0.7.

it is also possible to determine the polar

area moment of inertia by combining the "bottom" semicircle and the "upper" ellipse and afterwards subtracting the inner ellipses.

Additionally, the ability of a bone to resist stress was directly related to the cross-sectional

area moment of inertia, suggesting that as the area moment is reduced, the rate of rib stress fractures increases (1).

Therefore, a quick, simple way to compute or estimate Section Modulus (more specifically, its foundational parameter, Area Moment of Inertia) is needed so that we can move from sketch to improved sketch in our casting geometry brainstorming.

Interestingly, the difficulty in computing Area Moment of Inertia for casting shapes is one of the hidden reasons for the design and use of fabrications.

[Omega.sub.n] = [(Beta.sub.n.L).sup.2] [square root of [(EI/PL).sup.4]] where [Omega.sub.n] = the nth natural frequency; [Beta.sub.n] = a constant particular to the boundary conditions of the nth natural frequency; L = the length of the beam; E = the elastic modulus; I = the area moment of inertia; and P = the mass density of the system.

If a threaded rod is used for a spring, the root diameter should be used to calculate the area moment of inertia.

At marking the area moment of inertia of external diameter [d.sub.1] to upper side from axis x-x by letter [I.sub.1], moment of inertia of internal diameter [d.sub.2] by [I.sub.2], moments of inertia of cross-section [d.sub.1] and [d.sub.2] from axis x-x to axis of symmetry of these cross-sections by [I.sub.3] and [I.sub.4] accordingly, and area moments of inertia of lower halves of circles [d.sub.1] and [d.sub.2] by [I.sub.5] and [I.sub.6], one can see that common area moment of inertia of the first part of cross-section placed over the axis x-x is equal [I.sub.c1] = [I.sub.1] - [I.sub.2], while of the cross-section placed under the axis x-x is equal [I.sub.c2] = [I.sub.3] + [I.sub.5] - [I.sub.4] - [I.sub.6].

(1) it is possible to find the value [e.sub.1], necessary for the calculation of area moment of inertia of the neck cross-section.