User:EAS4200c.f08.Blue.E/Lecture 10

Homework Assignments

The first homework assigned in class consists of four individual problems.

The first Problem is to define the moment of inertia $$I_y$$ for a solid circular cross section through integration using polar coordinates. The integral being $$I_y=\int\int_A(Z^2dA)$$.

Also the derivation of $$I_y$$ for a square cross section is required which would be derived by the use of the integral $$I_y=\int\int_A(Z^2dYdZ)$$

The second problem assigned required comparing the moment of inertia defined for the circular cross section (case:1) in the previous problem to a redistributed equivalent area (case:2)

Where $$a=b$$, $$ A_2=3(a)(t)=\pi(r^2)$$,  and $$ t=a/10$$

The moment of inertia will be calculated assuming the y-axis passes through the centroid of case:1 and case:2 for fair comparison. The moment of inertia for case:2 will be calculated by using the fact that the moment of inertia for a rectagular cross section is $$I_y=1/12(b)(h^3)$$ along with the parallel axis theorem which is $$I_y=I+Ad^2$$

The third problem assigned was a modified version of problem 1.7. The modification was that all rectangular surfaces be changed to circular surfaces. The problem is to compare the moment of inertia $$I_y$$ of a circular cross section (case:1) to a modified cricular cross section (case:2) calculating the moment of inertia $$I_y$$ for both cases through the centroid of the cross section.

Where $$r_0=10cm$$, $$t=r_0/10$$, and $$A_1=A_2$$

The moment of inertia will be calculated for the circle from the answer to problem number one since it will be the same the moment of inertia for case 2 will be from a combination of rectangle moment of inertia and circular moment of inertia with use of the parallel axis theorem.

The fourth problem assigned was to show why a open rectangular cross section would be preferable to a hollow recangular cross section by comparing moment of inertia, manufacturing, and assembly techniques.