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单击以编辑母版标题样式,单击以编辑母版文本样式,第二级,第三级,第四级,第五级,*,Mechanics of Materials,Appendix,Geometric properties of planar graphs,Appendix Geometric properties of planar graphs,I-1,Static Moment and Centroid,I-2,Moment of Inertia and Radius of Inertia,I-3,Product of Inertia,I-5,Transfer,A,xis,F,ormula and,P,rincipal,I,nertia,A,xis,I-4,Parallel Axis Formula,Shearing stress on the cross section in torsion,I-1,Static Moment and Centroid,1,、,Static Moment,o,y,z,y,z,A,yd,A,A,zd,A,d,A,The static moment of,microarea for,z,axis,yd,A,The static moment of microarea for y,axis,zd,A,The static moment of the whole planar graph for,z,y,axises,Positive,negative,zero,y,z,o,dA,y,z,2.,F,ormula of centroid,c,oordinate,y,c,The static moment of the section for the centroidal axis is equal to zero.,If the static moment of any section for any axis is equal to zero,the axis passes the centroid,.,3,、,Static Moment and Centroid of the combined graphs,10,10,120,o,80,Solution,：,Divide the section into 1,，,2 parts,1,2,y,x,Sample 1 Try to determine the position of the centroid C.,10,10,120,o,80,1,2,y,x,Graph 1,Graph 2,So,10,10,120,o,80,1,2,y,x,I-2 Moment of Inertia and Radius of Inertia,o,y,z,y,z,d,A,1,、,Moment of inertia and radius of inertia,A,y,2,d,A,A,z,2,d,A,y,2,d,A,z,2,d,A,The moment of inertia of,microarea for,z,axis,The moment of inertia of,microarea for,y,axis,The moment of inertia,of the whole planar graph for,z,y,axises,Positive,o,y,z,y,z,d,A,The polar moment of inertia of planar graph for coordinate origin:,is the r,adius of,i,nertia,for y axis.,is the r,adius of,i,nertia,for x axis.,Sample 2:,Determine of the planar graph,y,z,b,h,z,dz,c,y,z,d,Sample 3:,Determine of the planar graph,d,D,y,z,Sample 4:,Determine of the planar graph,3,、,Moment of,i,nertia,of the combined graphs,b,h,y,Thinking:,o,y,z,y,z,d,A,y,z,When one of the two coordinate axis is symmetry axis，the product of inertia of the planar graph for the axis is equal to zero.,I-3,Product of Inertia,Positive,negative,zero,The,p,roduct of,i,nertia,of,planar graph,for,y,z,axis,y,c,z,c,y,c,z,c,O,Determine,:,Knowing：,Solution：,dA,C,0,I-4,Parallel Axis Formula,0,y,c,z,c,y,c,z,c,O,dA,C,0,Parallel,a,xis,f,ormula,Sample 5：Determine the,Moment of Inertia,of the graph for centroid axis,Yc,.,o,y,z,dA,y,z,I-5,Transfer,A,xis,F,ormula and,P,rincipal,I,nertia,A,xis,o,y,z,A,dA,y,z,Trigonometric function:,Sample 6,：,Determine,the moment of inertia and product of inerita for of the planar graph,Solution:,y,z,a,b,O,Discussion,：,When a is equal to b,，,the result?,P,rincipal,I,nertia,A,xis,?,Determine the two angles,Principal Inertia Axis:,Principal Inertia Moment:,Centroidal Principal Inertia,Ax,i,s,：,Centroidal Principal Inertia Moment,：,Formula of,Principal Inertia Moment,：,Sample 7,：,Determine the position of,centroidal,principal inertia,ax,i,s,and calculate,centroidal,principal inertia moment,120,80,10,10,1,、,Determine the,centroid,coordinate,z,y,O,C,z,C,y,C,2,、,Determin,the moment of inertia and product of inertia for,y,C,、,z,C,3,、,Determine the position of,centroidal,principal inertia axes and the,centroidal,principal,inertia moment,or,or,Centroidal Principal Inertia Axes,120,80,10,10,z,y,O,C,z,C,y,C,