Question 9.SP.4: The strength of a W14 × 38 rolled-steel beam is increased by......

MODELING and ANALYSIS: Place the origin O of coordinates at the centroid of the wide-flange shape, and compute the distance \bar{Y} to the centroid of the composite section by using the methods of Chap. 5 (Fig. 1). Refer to Fig. 9.13A for the area of the wide-flange shape. The area and the y coordinate of the centroid of the plate are

A=(9 \text { in. })(0.75 \text { in. })=6.75 ~\mathrm{in}^2

\bar{y}=\frac{1}{2}(14.1 \text { in. })+\frac{1}{2}(0.75 \text { in. })=7.425 \text { in. }

\bar{Y} \Sigma A=\Sigma \bar{y} A \quad \bar{Y}(17.95)=50.12 \quad \bar{Y}=2.792 ~\mathrm{in} .

Moment of Inertia. Use the parallel-axis theorem to determine the moments of inertia of the wide-flange shape and the plate with respect to the x’ axis. This axis is a centroidal axis for the composite section but not for either of the elements considered separately. You can obtain the value of \bar{I}_x for the wide-flange shape from Fig. 9.13A.

For the wide-flange shape,

I_{x^{\prime}}=\bar{I}_x+A \bar{Y}^2=385+(11.2)(2.792)^2=472.3 ~\mathrm{in}^4

For the plate,

I_{x^{\prime}}=\bar{I}_x+A d^2=\left(\frac{1}{12}\right)(9)\left(\frac{3}{4}\right)^3+(6.75)(7.425-2.792)^2=145.2 ~\mathrm{in}^4

For the composite area,

I_{x^{\prime}}=472.3+145.2=617.5 ~\mathrm{in}^4 \quad I_{x^{\prime}}=618 ~\mathrm{in}^4

Radius of Gyration. From the moment of inertia and area just calculated, you obtain

k_{x^{\prime}}^2=\frac{I_{x^{\prime}}}{A}=\frac{617.5 ~\mathrm{in}^4}{17.95 ~\mathrm{in}^2} \quad k_{x^{\prime}}=5.87 ~\mathrm{in}

REFLECT and THINK: This is a common type of calculation for many different situations. It is often helpful to list data in a table to keep track of the numbers and identify which data you need.