78 CE35-q.docx - CEE 213—Deformable Solids Please email [email protected] for the answer to the below question You can also email us

78 CE35-q.docx - CEE 213—Deformable Solids Please email...

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CEE 213—Deformable Solids Please email [email protected] for the answer to the below question. You can also email us for help on any of your assignments, testbanks etc. CEE 213—Deformable Solids CP 2 Properties of polygonal plane areas © Keith D. Hjelmstad 2014
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CEE 213—Deformable Solids CP 2—Properties of Areas Please email [email protected] for the answer to the below question. You can also email us for help on any of your assignments, testbanks etc. School for Sustainable Engineering and the Built Environment Ira A. Fulton Schools of Engineering Arizona State University The properties of cross sections. In the study of beams The properties of cross sections (including the axial bar and torsion) it becomes evident that much of the behavior is dictated by the properties of the cross sectional geometry . In fact, the resistance to deformation is always a function of the distribution of material in a cross section. For the axial bar the cross sectional area is key; for torsion the polar moment of inertia shows up; for flexure the moment of inertia about the axis of bending is an important property. These properties are essential to determining the deformation under load. Most of the classical “tricks of the trade” that can be used in service of the computation of geometric properties amount to making use of the properties of simple geometric pieces (often rectangles) to build up the overall cross sectional properties. © Keith D. Hjelmstad 2014 Cross Section x Longitudinal Axis z
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CEE 213—Deformable Solids CP 2—Properties of Areas What we show in this set of notes is that it is possible to derive the integrals needed to compute the cross sectional properties for a general triangle and then use that basic foundation to create a method to compute the cross sectional properties of any y closed polygonal region. This approach is based on two key observations: (1) all integrals can be divided into pieces and (2) it is possible to add in an integral that is not in the physical region as long as you subtract it right Cross Section back out. The key cross sectional properties. Consider the The key cross sectional properties general cross section defined at left. We establish a ( y , z ) coordinate system with origin O , and imagine another coordinate system ( x , h ) passing through the centroid C . For beam theory we need to compute area and moment of inertia (about the centroid). To compute the moment of inertia we need the location of the centroid. The area is simply defined as the sum of the infinitesimal areas that make up the cross section: The centroid of an area is defined as follows: © Keith D. Hjelmstad 2014 y z x h c ρ dA 2 e 3 e r O C A A dA 1 A dA A c r
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CEE 213—Deformable Solids CP 2—Properties of Areas In other words, it is the constant vector c that is the average over the area of the position r of the infinitesimal areas. Finally, the moment of inertia (tensor) is defined as Where the vector r is the distance from the centroid C to the
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