A vertical circular gate used to close the outlet of a large reservoir is submerged under water as shown: Since each product y 2 dA is positive, regardless of the sign of y, or zero (if y is zero), the integral I x will always be positive.Īnother example of a second moment, or moment of inertia, of an area is provided by the following problem from hydrostatics. It is obtained by multiplying each element of area dA by the square of its distance from the x axis and integrating over the beam section. In engineering practice, however, moment of inertia is used in commection with areas as well as masses. The term second moment is more proper than the term moment of inertia, since, logically, the latter should be used only to denote integrals of mass (include reference). The last integral is known as the second moment, or moment of inertia, of the beam section with respect to the x axis and is denoted by I x. The magnitude M of this couple (bending moment) must be equal to the sum of the moments Δ M x = y Δ F = ky 2 Δ A of the elemental forces. The system of forces Δ F thus reduces to a couple. The last integral obtained is recognized as the first moment Q x of the section about the x axis it is equal toĪ and is thus equal to zero, since the centroid of the section is located on the x axis. The magnitude of the resultant R of the elemental forces Δ F which act over the entire section is: The forces on one side of the neutral axis are forces of compression, while those on the other side are forces of tension on the neutral axis itself the forces are zero. Is known as the neutral axis of the section. This axis, represented by the x axis as shown: Vary linearly with the distance y between the element of area Δ A and an axis passing through the centroid of the section. Such a beam is said to be in pure bending, and it is shown in mechanics of materials that the internal forces in any section of the beam are distributed forces whose magnitudes: Second Moment, or Moment of Inertia, of an AreaĬonsider a beam of uniform cross section which is subjected to two equal and opposite couples applied at each end of the beam. EngArc - L - Second Moment, or Moment of Inertia, of an Area
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |