Cantilever Beam Elastic Deflection & Tangent Stiffness (Civil Engineering)

A civil structural engineer is designing a steel cantilever balcony projection for a multi-story casino resort on the Las Vegas Strip. Under a uniform dead load $w_0$ (in $\text{kN/m}$), the downward vertical elastic deflection curve $y(x)$ along the length of the beam ($0 \le x \le L$) is governed by the fourth-degree polynomial:

where $E$ is the elastic modulus of structural steel ($200\text{ GPa}$), $I$ is the cross-sectional moment of inertia ($4.5 \times 10^{-5}\text{ m}^4$), $L = 6.0\text{ m}$ is the total cantilever span, and $x$ is the horizontal distance measured from the rigid wall support ($x = 0$) to the free tip ($x = L$).

1. Compute the first derivative $\theta(x) = \frac{dy}{dx}$ of the deflection curve using the Power and Constant Multiple Rules to determine the rotational slope of the beam.
2. Verify that the fixed clamp condition at the wall support ($x = 0$) satisfies $\theta(0) = 0$, and compute the exact slope of the beam at the free cantilever tip ($x = L$).
3. Formulate the equation of the tangent line to the deflection curve at the mid-span location $x = \frac{L}{2} = 3.0\text{ m}$ in terms of the load parameter $C = \frac{w_0}{24EI}$, and explain how structural engineers use tangent slopes to ensure glass curtain-wall facade panels do not bind or shatter under building sway.