Structural Beam Bending Moment Optimization (Civil & Construction Engineering)

In civil and construction engineering, horizontal reinforced concrete beams and steel girders spanning bridge roadways must withstand internal bending stresses induced by combined self-weight and dynamic traffic loads. Under standard beam bending theory, internal flexural fiber stress $\sigma(x)$ is directly proportional to the internal bending moment $M(x)$ across the beam cross-section:

where $y$ is the distance from the neutral axis and $I$ is the cross-sectional area moment of inertia. Structural failure (cracking or tensile rupture) initiates at the exact point along the span where the bending moment reaches its absolute peak.

A structural engineer evaluates a 6-meter long horizontal reinforced support beam across a closed building bay. The internal bending moment $M$ (in kilonewton-meters, $\text{kN}\cdot\text{m}$) at any horizontal position $x$ (in meters from the left support wall) over the closed span $x \in [0, 6]$ is accurately modeled by:

1. Verify that the hypotheses of the Extreme Value Theorem are satisfied for $M(x)$ on $[0, 6]$.
2. Compute $M'(x)$ (the internal shear force distribution $V(x)$) and locate all interior critical numbers within $(0, 6)$.
3. Execute the 5-step Closed Interval Method algorithm to determine the absolute maximum bending moment and absolute minimum bending moment, identifying both the extreme values and their exact longitudinal positions.
4. Explain how the structural design engineer uses the calculated peak bending moment location to optimize steel rebar placement, detailing why placing heavy reinforcement uniformly across the entire 6-meter span would constitute costly construction waste.