10. Moments of inertia

• Develop a method to determine the moment of inertia for an area• Introduce the product of inertia andshow how to determine max and minmoments of inertia for an area• Discuss mass moment of inertia

10.1 Definition of moments of inertia for areas

Whenever a distributed loading acts perpendicular to an area and its intensity varies linearly, the computation of the moment of the loading distribution about an axis will involve a quantity called the moment of inertia of the area

10.1 Definition of moments of inertia for areas

• The plate is subjected to a fluid pressure p• Pressure varies linearly with depth; p = γy where γ is specific weight of fluid• The force acting on the differential area dA of the plate is dF = p dA and dF = γy dA• The moment of dF about the x axis is dM = y dF = γy2 dA and so integrating dM over the entire area of the plate yields;

The moment of inertia Ix

of the area about x-axis

Moment of inertia

For the entire area A the moments of inertia are determined by integration;

Polar moment of inertia about the‘pole’ O or z axis;

10.2 Parallel axis theorem for an area

The parallel –axis theorem can be used to find the moment of inertia of an area about any axis that is parallel to an axis passing through the centroid and about which the moment of inertia is known

10.2 Parallel axis theorem for an area

Polar momentof inertia

Moment of inertia of the areaabout the centroidal axis

The moment of inertia for an area about an axis is equal to its moment of inertia about a parallel axis passing through the area’s centroid plus the product of the area and the square of the perpendicular distance between the axes

10.3 Radius of gyration of an area

• The radius of gyration of an area about an axis has units of length andis a quantity that is often used for the design of columns in structural Mechanics• If the areas and moments of inertia are known, the radii of gyration, k, are determined from the formulas

This week’s schedule

Tuesday: Final reviewWednesday: Final exam

Thursday: Truss project presentations (5 min each team)