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2009+7-1InternalForces

2009+7-1InternalForces - Introduction to Solid...

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1 SJTU Introduction to Solid Mechanics -Vm211 MOMENTS OF INERTIA = A A dA dA x x dA x = A y dA x I dA x 2 2 x x dA C ˉ x First moment of the area Second moment of the area SJTU Introduction to Solid Mechanics -Vm211 MoI – DEFINITION For the differential area dA, shown in the figure: d I x = y 2 dA , d I y = x 2 dA , d J O = r 2 dA , where J O is the polar moment of inertia about the pole O or z axis. SJTU Introduction to Solid Mechanics -Vm211 MoI – DEFINITION The moments of inertia for the entire area are obtained by integration. I x = A y 2 dA ; I y = A x 2 dA J O = A r 2 dA = A ( x 2 + y 2 ) dA = I x + I y where J O is the polar moment of inertia about the pole O or z axis SJTU Introduction to Solid Mechanics -Vm211 MoI – DEFINITION The moments of inertia for the entire area are obtained by integration. I x = A y 2 dA ; I y = A x 2 dA J O = A r 2 dA = A ( x 2 + y 2 ) dA = I x + I y The MoI is also referred to as the second moment of an area and has units of length to the fourth power (m 4 ) . SJTU Introduction to Solid Mechanics -Vm211 RADIUS OF GYRATION OF AN AREA For a given area A and its MoI, I x , imagine that the entire area is located at distance k x from the x axis. 2 Then, I x = k x A or k x = ( I x / A). This k x is called the radius of gyration of the area about the x axis. Similarly; k Y = ( I y / A ) and k O = ( J O / A ) A k x x y SJTU Introduction to Solid Mechanics -Vm211 RADIUS OF GYRATION OF AN AREA The radius of gyration has units of length and gives an indication of the spread of the area from the axes. A k x x y

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2 SJTU Introduction to Solid Mechanics -Vm211 MoI FOR AN AREA BY INTEGRATION For simplicity, the area element used has a differential size in only one direction (dx or dy). This results in a single integration and is usually simpler than doing a double integration with two differentials, dx·dy. SJTU Introduction to Solid Mechanics -Vm211 The step-by-step procedure is: 1. Choose the element dA: There are two choices: a vertical strip or a horizontal strip. Some considerations about this choice are: a) The element parallel to the axis about which the MoI is to be determined usually results in an easier solution. For example, we typically choose a vertical strip for determining I y . MoI BY INTEGRATION SJTU Introduction to Solid Mechanics -Vm211 b) If y is easily expressed in terms of x (e.g., y = x 2 + 1), then choosing a vertical strip with a differential element dx wide may be advantageous. MoI BY INTEGRATION SJTU Introduction to Solid Mechanics -Vm211 MoI BY INTEGRATION 2. Integrate to find the MoI. For example, given the element shown in the figure above: I y = x 2 dA = x 2 y dx and I x = d I x = (1 / 3) y 3 dx ( using the information for a rectangle about its base from the inside back cover of the textbook ).
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2009+7-1InternalForces - Introduction to Solid...

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