东南大学：《建筑力学 Architectural Mechanics》课程教学课件（英文讲稿）A06 Centroids and Centers of Gravity

Centroids & Centers of Gravity

Centroids & Centers of Gravity

Contents·Introduction（绪论）·CenterofGravityofa2DBody（两维物体的重心）·Centroids and First Moments of Areas（形心与面积的一次矩）·CentroidsofCompositeAreas（组合面积的形心）·Determination ofCentroids by Integration（积分法求形心）·CentroidsofCommonShapesofAreas（常见平面图形的形心）2

Contents • Introduction（绪论） • Center of Gravity of a 2D Body（两维物体的重心） • Centroids and First Moments of Areas（形心与面积的一次矩） • Centroids of Composite Areas（组合面积的形心） • Determination of Centroids by Integration（积分法求形心） • Centroids of Common Shapes of Areas（常见平面图形的形心） 2

Lntroduction. The earth exerts a gravitational force on each of the particlesforming a body. These forces can be replaced by a singleequivalent force equal to the weight of the body and applied at thecenter of gravity for the body. The centroid of an area is analogous to the center of gravity of abody. The concept of the first moment of an area is used to locatethe centroid.3

Introduction • The earth exerts a gravitational force on each of the particles forming a body. These forces can be replaced by a single equivalent force equal to the weight of the body and applied at the center of gravity for the body. • The centroid of an area is analogous to the center of gravity of a body. The concept of the first moment of an area is used to locate the centroid. 3

Center of Gravity of a 2D Body.Center of gravity of a plateAZ,=x=xa=J xdWZM,=JW-y,AW=J ydw4

Center of Gravity of a 2D Body • Center of gravity of a plate y i i x i i M xW x W xdW M yW y W ydW               4

Centroids and First Moments of Areas.Centroid ofan area (assuminguniformthicknessanddensity)xW=[xdWx(pgAt)=[x(pgt)dAxA=[xdA=S,= first moment with respect to yyA=[ydA=Sx= first moment with respectto x5

Centroids and First Moments of Areas     first moment with respect to first moment with respect to y x xW x dW x gAt x gt dA xA x dA S y yA y dA S x               • Centroid of an area (assuming uniform thickness and density) 5

Centroids and First Moments of AreasB. An area is symmetric with respect to an axis BBif for every point P there exists a point P'suchthat PP'is perpendicular to BB'and is dividedinto two equal parts by BB'.B(a).Thefirstmoment ofanareawithrespecttoaline of symmetry is zero..If an area possesses a line of symmetry,itscentroid lies on that axis. If an area possesses two lines of symmetry, itsBcentroid lies at their intersection..An area is symmetric with respect to a center Oif for every element dA at (x,y) there exists anarea dA' of equal area at (-x,-y). The centroid of the area coincides with thecenterof symmetry6

Centroids and First Moments of Areas • An area is symmetric with respect to an axis BB’ if for every point P there exists a point P’ such that PP’ is perpendicular to BB’ and is divided into two equal parts by BB’. • The first moment of an area with respect to a line of symmetry is zero. • If an area possesses a line of symmetry, its centroid lies on that axis • If an area possesses two lines of symmetry, its centroid lies at their intersection. • An area is symmetric with respect to a center O if for every element dA at (x,y) there exists an area dA’ of equal area at (-x,-y). • The centroid of the area coincides with the center of symmetry. 6

Centroids of Composite Areas·CompositeplatesW3XEW=Ex,WWWYw=Zyw,G3TyEAsCompositeareaC3AZAXZA=ZXAAXYZA-ZJAI7COO7

Centroids of Composite Areas • Composite plates i i i i X W xW Y W yW       • Composite area i i i i X A x A Y A y A       7

Sample ProblemS.=xdA= Ax( ydA = AyS,=S,-ZSx=ZA ； S,=-Z4x,100i=1i=1yOiC1120一ZA,x,S10×2000 +(20 + 70)×2800i=lx :AA2000+2800= 56.66140n≥4J.S正i=l=50yAA---12X-8

1 1 1 ; n n n x xi i i y i i i i S S A y S A x           1 1 10 2000 20 70 2800 2000 2800 56.66 50 n i i y i n i i x i A x S x A A A y S y A A                 x y A A S ydA Ay S xdA Ax       O x 100 y 20 20 140 C C1 C2 Ⅰ Ⅱ Sample Problem 8

Sample ProblemSOLUTION:y120mm: Divide the area into a triangle, rectangleand semicircle with a circular cutout.60mm40mm.Calculate the first moments of eacharea80mmwithrespecttothe axes.x.Findthetotal areaandfirstmoments of60mm+the triangle, rectangle, and semicircle.Subtract the area and first moment of thecircularcutout.For the plane area shown, determinethe first moments withrespect to the·Compute the coordinates of the areax and y axes and the location of thecentroid by dividingthe first moments bycentroid.the total area.9

For the plane area shown, determine the first moments with respect to the x and y axes and the location of the centroid. SOLUTION: • Divide the area into a triangle, rectangle, and semicircle with a circular cutout. • Compute the coordinates of the area centroid by dividing the first moments by the total area. • Find the total area and first moments of the triangle, rectangle, and semicircle. Subtract the area and first moment of the circular cutout. • Calculate the first moments of each area with respect to the axes. Sample Problem 9

yy4r1=25.46mm120mmri=60mm3元ri=40mm=60mm60mm十+rg=40mm40mm105.46mm80mm80mm80mm140mmxxxx60mm60mm60mm20mmyA,mm3xA, mm3A, mm?x, mmy, mmComponent+384×10360+576×103(120)(80)=9.6×10340Rectangle-72×103+144×103（120)(60)=3.6×10340-20Triangle+596.4×103+339.3×103元(60)2=5.655×10360105.46Semicircle-402.2×103-301.6×1036080-(40)2=-5.027×103CircleZyA=+506.2×103XA=+757.7×103ZA=13.828×103S, = +506.2×103mm3Find the total area and first moments of thetriangle,rectangle,and semicircle.SubtracttheS, = +757.7×103mm3areaandfirstmomentofthecircularcutout10

3 3 3 3 506.2 10 mm 757.7 10 mm x y S S       • Find the total area and first moments of the triangle, rectangle, and semicircle. Subtract the area and first moment of the circular cutout. 10