Structural Stress Analysis in Spherical Joints Solution

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.

Step-by-Step Solution:

1. State the Domain Condition and Classify 3D Volume: State the Radicand Non-Negativity Condition for Three-Variable Functions:

   $$25 - (x^2 + y^2 + z^2) \ge 0$$
   Rearranging terms:
   $$x^2 + y^2 + z^2 \le 25$$
   • Set-Builder Notation:
     $$D = \{(x, y, z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 \le 25\}$$
   • Geometric Description: The domain represents a solid 3D sphere (a closed ball) centered at the origin $(0, 0, 0)$ with a boundary radius of $R = \sqrt{25} = 5\text{ cm}$.
   • Topological Classification: Because the domain includes its outer spherical boundary surface $x^2 + y^2 + z^2 = 25$, it is a closed and bounded solid volume in $\mathbb{R}^3$. (Defined in Domain and Range of Multivariable Functions.)

2. Determine the Stress Range: State the Three-Variable Quadratic Extremum Formulation: The radial distance squared from the center is $\rho^2 = x^2 + y^2 + z^2 \ge 0$.

   • Maximum Stress ($\rho = 0$): At the geometric center core $(0, 0, 0)$, $\rho^2 = 0$:
     $$\sigma_{\max} = h(0, 0, 0) = 50\sqrt{25 - 0} = 50(5) = 250\text{ MPa}$$
   • Minimum Stress ($\rho = 5$): Along the entire outer spherical shell $x^2 + y^2 + z^2 = 25$:
     $$\sigma_{\min} = 50\sqrt{25 - 25} = 0\text{ MPa}$$
   • Range in Interval Notation:
     $$R = [0, 250]\text{ MPa}$$

3. Evaluate Inspection Coordinate Point $(3, 4, 2)$: Compute the distance squared from the origin to the inspection point:

   $$\rho^2 = x^2 + y^2 + z^2 = 3^2 + 4^2 + 2^2 = 9 + 16 + 4 = 29$$
   Test the domain inequality:
   $$29 \le 25 \quad \text{(False, since } 29 > 25\text{)}$$
   Substituting into the mathematical function yields:
   $$\sigma = 50\sqrt{25 - 29} = 50\sqrt{-4} = 100i \notin \mathbb{R}$$
   The point $(3, 4, 2)$ is located at distance $\rho = \sqrt{29} \approx 5.385\text{ cm} > 5\text{ cm}$ from the center, placing it strictly outside the steel joint.

Meaning of the Answer & Real-Life Application:

The point $(3, 4, 2)$ lies in ambient open air $0.385\text{ cm}$ beyond the spherical metal casing. Evaluating the stress formula generates an imaginary number, signifying that the physical continuum model is non-existent beyond the outer steel surface. In automated NDE robotics, domain-checking algorithms prevent scanner arms from hunting for internal echoes outside the workpiece boundary, preventing false defect alerts and calibration faults during structural inspections.