chapter10 - MOMENTS OF INERTIA (Chapter 10) APPLICATIONS...

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MOMENTS OF INERTIA (Chapter 10)
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APPLICATIONS Many structural members like beams and columns have cross sectional shapes like I, H, C, etc. . Why do they usually not have solid rectangular, square, or circular cross sectional areas? What primary property of these members influences design decisions? How can we calculate this property?
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APPLICATIONS (continued) Many structural members are made of tubes rather than solid squares or rounds. Why? What parameters of the cross sectional area influence the designer’s selection? How can we determine the value of these parameters for a given area?
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THE CONCEPT OF THE MoI OF AN AREA Consider a plate submerged in a liquid. The pressure of a liquid at a distance z below the surface is given by p = γ z, where γ is the specific weight of the liquid. The force on the area dA at that point is dF = p dA. The moment about the x-axis due to this force is z (dF). The total moment is A z dF = A γ z 2 dA = γ A ( z 2 dA). This sort of integral term also appears in solid mechanics when determining stresses and deflection. This integral term is referred to as the moment of inertia of the area of the plate about an axis.
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THE CONCEPT OF THE MoI (continued) Consider three different possible cross sectional shapes and areas for the beam RS. All have the same total area and, assuming they are made of same material, they will have the same mass per unit length. For the given vertical loading P on the beam, which shape will develop less internal stress and deflection? Why? The answer depends on the MoI of the beam about the x-axis. It turns out that Section A has the highest MoI because most of the area is farthest from the x axis. Hence, it has the least stress and deflection. 10cm 10cm 1cm 1cm x 3cm 10cm 3cm R S P (C) (B) (A)
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MoI DEFINITION For the differential area dA, shown in the figure: d I x = y 2 dA , d I y = x 2 dA , and, d J O = r 2 dA , where J O is the polar moment of inertia about the pole O or z
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This note was uploaded on 04/20/2011 for the course ENG 1333 taught by Professor Brr during the Spring '10 term at American College of Computer & Information Sciences.

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chapter10 - MOMENTS OF INERTIA (Chapter 10) APPLICATIONS...

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