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Tony's Intertwingled Memex • Reference Manual

Cantilever Beam Elastic Deflection & Tangent Stiffness (Civil Engineering)

Formulas & Definitions

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 w0w_0 (in kN/m\text{kN/m}), the downward vertical elastic deflection curve y(x)y(x) along the length of the beam (0≤x≤L0 \le x \le L) is governed by the fourth-degree polynomial:

y(x)=w024EI(x4−4Lx3+6L2x2)y(x) = \frac{w_0}{24EI}\left(x^4 - 4Lx^3 + 6L^2x^2\right)
where EE is the elastic modulus of structural steel (200 GPa200\text{ GPa}), II is the cross-sectional moment of inertia (4.5×10−5 m44.5 \times 10^{-5}\text{ m}^4), L=6.0 mL = 6.0\text{ m} is the total cantilever span, and xx is the horizontal distance measured from the rigid wall support (x=0x = 0) to the free tip (x=Lx = L).

  1. Compute the first derivative θ(x)=dydx\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=0x = 0) satisfies θ(0)=0\theta(0) = 0, and compute the exact slope of the beam at the free cantilever tip (x=Lx = L).
  3. Formulate the equation of the tangent line to the deflection curve at the mid-span location x=L2=3.0 mx = \frac{L}{2} = 3.0\text{ m} in terms of the load parameter C=w024EIC = \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.