Civil Structural Girder Web Plate Centroid Analysis (Civil & Construction Engineering)

A structural engineer and construction project manager at the UNLV Howard R. Hughes College of Engineering are detailing custom fabricated steel gusset connection plates for an elevated highway interchange bridge.

The planar gusset plate occupies a general region $R$ in the $xy$ -plane bounded below by the parabolic profile $y = \frac{1}{2}x^2$ and above by the horizontal top flange line $y = 2$ (with spatial coordinates measured in meters):

R = \left\{(x, y) \in \mathbb{R}^2 \;\middle|\; -2 \le x \le 2, \, \frac{1}{2}x^2 \le y \le 2\right\}

To balance structural dead loads and prevent secondary eccentric twisting torques when heavy construction cranes lift and bolt the plate into position, the engineering team must determine the vertical centroid coordinate $\bar{y}$ using the standard mechanics of materials formulation:

\bar{y} = \frac{\iint_R y \, dA}{\iint_R 1 \, dA}

Determine the outer constant limits of integration by equating the boundary functions and finding their real intersection points.
Formulate and evaluate the denominator integral $A = \iint_R 1 \, dA$ representing the geometric planar area of the steel gusset plate.
Formulate and evaluate the numerator integral $M_x = \iint_R y \, dA$ representing the first moment of area with respect to the $x$ -axis.
Calculate the vertical centroid coordinate $\bar{y}$ in meters and explain why $\bar{y} > 1.0\text{ m}$ .