Structural Beam Bending & Stress Inversion (Mechanical Engineering)

In solid mechanics and structural engineering, statically indeterminate structural beams subjected to asymmetric lateral and point loads develop internal bending moments $M(x)$ that vary along the beam's longitudinal span. Under Euler-Bernoulli beam theory, the vertical deflection curve $w(x)$ is related to the bending moment by $M(x) = E I w''(x)$, where $E$ is Young's modulus and $I$ is the area moment of inertia.

A mechanical engineer is analyzing an overhead gantry crane support girder of length $L = 6\text{ meters}$. Under operating crane carriage loads, the internal bending moment distribution $M(x)$ (in $\text{kN}\cdot\text{m}$) as a function of position $x \in [0, 6]$ along the girder is modeled by the cubic polynomial:

1. Find the roots of $M(x) = 0$ along the girder span $[0, 6]$ to locate the neutral load pinning stations ($x$-intercepts).
2. Calculate the first derivative $M'(x)$ (representing internal shear force $V(x) = \frac{dM}{dx}$), locate all stationary critical points, and classify local peak and valley bending moments using the First Derivative Test.
3. Compute the second derivative $M''(x)$, determine the intervals of concavity, and find the coordinates of all points of inflection.
4. In structural design, a point of inflection in the deflection curve ($w''(x) = 0 \iff M(x) = 0$) marks a point of contraflexure where internal stresses invert between upper-flange compression and upper-flange tension. Explain why identifying inflection points in $M(x)$ governs the optimal placement of welded steel web stiffeners to prevent localized shear buckling.