Mae 241-lec15
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10. Moments of inertia
• Develop a method to determine the moment of inertia for an area • Introduce the product of inertia and show how to determine max and min moments 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 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
Polar moment of inertia
Moment of inertia of the area about the centroidal axis
10.3 Radius of gyration of an area
• The radius of gyration of an area about an axis has units of length and is 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 review Wednesday: Final exam Thursday: Truss project presentations (5 min each team)
• Develop a method to determine the moment of inertia for an area • Introduce the product of inertia and show how to determine max and min moments 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 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
Polar moment of inertia
Moment of inertia of the area about the centroidal axis
10.3 Radius of gyration of an area
• The radius of gyration of an area about an axis has units of length and is 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 review Wednesday: Final exam Thursday: Truss project presentations (5 min each team)
