Structural Beam Bending Moment Optimization (Civil & Construction Engineering) Solution

Problem Overview

We analyze the internal bending moment distribution $M(x) = -2x^3 + 15x^2 - 24x + 50$ ($\text{kN}\cdot\text{m}$) across a 6-meter span ($x \in [0, 6]$). We verify the Extreme Value Theorem, locate critical points, execute the Closed Interval Method, and determine structural steel rebar placement to maximize safety while minimizing material costs.

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Step-by-Step Mathematical Solution

Step 1: Verify Extreme Value Theorem Hypotheses

The governing theorem is the Extreme Value Theorem (EVT):

A continuous function on a closed interval $[a, b]$ is guaranteed to achieve an absolute maximum and an absolute minimum. (This theorem is defined in Extreme Value Theorem.)

1. Continuity: $M(x)$ is a cubic polynomial. Polynomials are continuous for all real numbers $\mathbb{R}$; hence $M(x)$ is continuous on $[0, 6]$.
2. Closed Domain: The domain $[0, 6]$ explicitly includes both structural support wall boundaries ($x = 0$ and $x = 6$).
3. Conclusion: Both hypotheses are satisfied. Absolute extrema are guaranteed to exist.

Step 2: Differentiate and Locate Interior Critical Numbers

The governing concept is Critical Numbers:

(This concept is defined in Critical Numbers and Critical Points.)

1. Differentiate using the Power Rule:
   $$M'(x) = \frac{d}{dx}[-2x^3 + 15x^2 - 24x + 50] = -6x^2 + 30x - 24$$
   (In beam mechanics, the derivative of the bending moment is the internal shear force: $V(x) = \frac{dM}{dx}$. Thus, critical points of the bending moment occur precisely where shear force is zero).
2. Set $M'(x) = 0$ to find stationary points:
   $$-6(x^2 - 5x + 4) = 0$$
   Factoring the quadratic:
   $$-6(x - 1)(x - 4) = 0 \implies x = 1\text{ m} \quad \text{and} \quad x = 4\text{ m}$$
3. Check singular points: $M'(x)$ is a polynomial, defined everywhere.
4. Candidate Filtering: Both $x = 1\text{ m}$ and $x = 4\text{ m}$ lie strictly within the interior interval $(0, 6)$. Both are retained.

Step 3: Execute Candidate Height Evaluation (Closed Interval Method)

The governing algorithm is the Closed Interval Method: Evaluate the original bending moment function $M(x)$ at all surviving interior critical numbers and boundary endpoints:

1. Left Support Wall ($x = 0\text{ m}$):
   $$M(0) = -2(0)^3 + 15(0)^2 - 24(0) + 50 = 50\text{ kN}\cdot\text{m}$$
2. First Critical Point ($x = 1\text{ m}$):
   $$M(1) = -2(1)^3 + 15(1)^2 - 24(1) + 50 = -2 + 15 - 24 + 50 = 39\text{ kN}\cdot\text{m}$$
3. Second Critical Point ($x = 4\text{ m}$):
   $$M(4) = -2(4)^3 + 15(4)^2 - 24(4) + 50 = -2(64) + 15(16) - 96 + 50$$
   $$M(4) = -128 + 240 - 96 + 50 = 66\text{ kN}\cdot\text{m}$$
4. Right Support Wall ($x = 6\text{ m}$):
   $$M(6) = -2(6)^3 + 15(6)^2 - 24(6) + 50 = -2(216) + 15(36) - 144 + 50$$
   $$M(6) = -432 + 540 - 144 + 50 = 14\text{ kN}\cdot\text{m}$$

Step 4: Compare Heights and Identify Extrema

Comparing the four candidate values $\{50, 39, 66, 14\}$:

• Absolute Maximum Bending Moment Value: $66\text{ kN}\cdot\text{m}$ at location $x = 4\text{ meters}$.
• Absolute Minimum Bending Moment Value: $14\text{ kN}\cdot\text{m}$ at location $x = 6\text{ meters}$.

Step 5: Civil Engineering Design Optimization

1. Structural Safety and Rebar Clustering: The absolute peak bending moment of $66\text{ kN}\cdot\text{m}$ represents the point of maximum tensile flexural stress in the concrete beam fibers. If structural failure occurs under dynamic loading, cracking will initiate at the bottom tensile face at $x = 4\text{ meters}$. The civil engineer uses this calculus calculation to specify dense groupings of high-yield steel rebar stirrups and longitudinal tension bars concentrated in the zone around $x = 4\text{ meters}$.
2. Material Cost Savings vs. Uniform Sizing: At the right support wall ($x = 6\text{ meters}$), the internal bending moment drops to its minimum of $14\text{ kN}\cdot\text{m}$—nearly 80% lower than the maximum. If an engineer lazily specified uniform maximum rebar reinforcement across the entire 6-meter span, the project would waste tons of expensive steel reinforcement in low-stress zones. By tapering rebar density according to the moment profile, construction costs are minimized while satisfying structural building codes.