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1、Chapter 12 Review of Centroids and Moments of Inertia Mechanics of MaterialsCentroids of Plane AreasWelcome to mechancis of materials,in this video,we are going to discuss how to locate centroids of plane Areas.Centroids of Plane Areas1.The area of the geometric figure:dA:a differential element of a
2、rea;x and y:coordinates;A:the total area of the figure.2.The first moments of the area:=The position of the centroid of a plane area is an important geometric property.Units:m3,mm3Units:m2,mm2The position of the centroid of a plane area is an important geometric property.To obtain formulas for locat
3、ing centroids,lets refer to this Figure,which shows a plane area of irregular shape with its centroid at point C.The xy coordinate system is oriented arbitrarily with its origin at any point O.The area of the geometric figure is defined by the following integral.and units of area is length raised to
4、 the second power;for instance,m2 or mm2The first moments of the are represent the sums of the products of the differential areas and their coordinates.The first moments with respect to the x and y axes are defined,respectively,as follows.First moments may be positive or negative,depending upon the
5、position of the xy axes.Also,first moments have units of length raised to the third power;for instance,m3 or mm3symmetric about an axis two axes of symmetry symmetric about a point center of symmetry Boundaries of the area are defined by simple mathematical expressions:Specical cases 3.The centroid
6、of a plane area:-locate the centroid by evaluating the integralsThe coordinates x and y of the centroid C are equal to the first moments divided by the area.If the boundaries of the area are defined by simple mathematical expressions,we can evaluate the integrals appearing in these two equations in
7、closed form and thereby obtain formulas for x and y.Now lets look at some special cases.First case,a singly symmetric area,the centroid must lie on the x axis,which is the axis of symmetry.Therefore,only one coordinate needed be calculated in order to locate the centroid C.In general,if an area is s
8、ymmetric about an axis,the centroid must lie on that axis because the first moment about an axis of symmetry equals zero.Second case,if an area has two axes of symmetry,as illustrated in the figure,theposition of the centroid can be determined by inspection,because it lies at the intersection of the
9、 axes of symmetry.third case,area symmetric about a point.The area has no axes of symmetry,but every line drawn through that point contacts the area in a symmetrical manner.This point is called the center of symmetry.And the centroid of such an area coincides with this point.Therefore the centroid c
10、an be located by inspection.xi,yi:the coordinates of the centroid of the ith element Irregular boundaries not defined by simple mathematical expressions:-locate the centroid by numerically evaluating the integrals.n:the total number of elementsCoordinates of centroid C:The first moments:The total ar
11、ea:Note:The accuracy of the calculations depends upon how closely the selected elements fit the actual area.If an area has irregular boundaries not defined by simple mathematical expressions,we can locate the centroid by numerically evaluating the integrals.The simplest procedure is to divide the ge
12、ometric figure into small finite elements and replace the integrations with summations.After divided into small elements,the total area A of the figure is the summation of deltaAi,i from 1 to n,n is the total number of elements and delta Ai represents the area of the i th element.The first moment of
13、 the area about x axis Qx is equal to the summation of yi times deltaAi,i from 1 to n.yi is the y coordinate of the centroid of the ith element.And same for jthe first moment of the area about y axis Qy.Replacing the integrals by the corresponding summations,the formulas for the coordinates of the c
14、entroid are.note,the accuracy of the calculations for the coordinates of centroid C depends upon how closely the selected elements fit the actual area.If they fit exactly,the results are exact.Example 1Solution:Area of the strip element:Total area of this figure:To determine the coordinates x and y
15、of the centroid C,we will use the integrations.First,selecting an element of area dA in theform of a thin vertical strip of width dx and height y.The area of this differential element dA is .Therefore,the area of the parabolic semisegment is equal to the integration of dA,equal to 2/3 bh,Note that t
16、his area is 2/3 of the area of the surrounding rectangle.Question:Is it possible to select a horizontal strip element to locate the centroid C of the parabolic semisegment?If possible,please try this way to find the coordinates of the centroid C.The first moments Qx and Qy:The coordinates of the cen
17、troid C:For the strip element dA,the distance from its centroid to x axis is*y/2,therefore the first moment of the parabolic semisegment with respect x axis*Qx is equal to the integration of y/2 times the area dA,*equal to 4bh2/15.Similarly,we can calculate *Qy.Then we can now determine the coordinates of the centroid C*.Question:Is it possible to select a horizontal strip element to locate the centroid C of the parabolic semisegment?If possible,please try this way to find the coordinates of the centroid C.This is end of this video.End