STATICS  CASE STUDY SOLUTION

Rail Subparts
Rail Dimensions


The beam cross section can be broken into 4 composite parts consisting of 3 rectangles and a triangle as shown. Orient an axis system as shown, so that the cross section lies entirely in the xy plane. Since the thickness is constant for all the parts, the centroid can be found using the area equations,



Composite Part 1

Part 1


Part 1 is a rectangle with an area of
A_{1} = (2)(3) = 6 cm^{2}
The centroid
of a rectangle lies at half its width and half its height,
so for part 1



Composite Part 2

Part 2


Part 2 is a rectangle with an area of
A_{2} = (16)(2) = 32 cm^{2}
Its centroid is located at



Composite Part 3

Part 3


Part 3 is a rectangle with an area of
A_{3} = (2)(10) = 20 cm^{2}
Its centroid is located at



Composite Part 4

Part 4


Part 4 is a triangle with an area of
A_{4} = (0.5)(4)(4) = 8 cm^{2}
The
centroid of a right triangle is located twothirds of the distance from the vertex to the other end; therefore, for part 4



Centroid of Total System

Total System


With these results, find the total area of the system is
Substituting the areas and centroid locations for each of the individual parts into the first two equations gives



