Structural Stress Analysis in Spherical Joints

In civil and mechanical structural engineering, large-span space frame roofs and pedestrian bridges rely on solid spherical cast-steel node joints to anchor converging diagonal structural struts. Under combined tension, compression, and shear loads from bridge traffic and wind gusts, a 3D internal stress tensor field develops inside each spherical node.

A structural engineering research team at UNLV models the equivalent Von Mises stress distribution $\sigma$ (in megapascals, $\text{MPa}$) inside a solid spherical steel support joint of radius $R = 5\text{ cm}$ using the three-variable radical function:

where $x, y, z$ represent spatial coordinates (in centimeters) measured from the geometric center core of the node $(0, 0, 0)$.

An ultrasonic nondestructive evaluation (NDE) transducer inspects internal material properties at the coordinate point $(3, 4, 2)$.

1. State the algebraic domain condition for $h(x, y, z)$ and write the mathematical domain $D$ in set-builder notation. Describe the geometric 3D solid region it represents.
2. Determine the mathematical range $R$ of the stress function in interval notation. Identify the coordinates where maximum stress occurs and where stress drops to zero.
3. Evaluate whether the inspection point $(3, 4, 2)$ lies within the physical material domain of the steel joint. Explain the engineering significance of this result for structural inspection workflows.