截面的几何性质面积矩惯性矩惯性积平行移轴公式

HOHAIUNIVERSITY1附录A截面的几何性质§A-1截面的面积矩和形心位置一、面积矩的定义yzodAzySy=zdA∫A∫ASz=ydA面积矩可为正、负或为零。HOHAIUNIVERSITY2二、截面形心的位置yc=∫AydAA=SzAzc=∫AzdAA=SyAyzoCdAzczycy故Sz=AycSy=Azc形心轴：过平面图形形心的轴截面对形心轴的面积矩为零。HOHAIUNIVERSITY3例1求如图矩形Sz和Syb/2b/2h/2h/2zydadyy解：AzAySd)2(habhCyA同样地CyzAbdbhS)2(haayybdHOHAIUNIVERSITY4三、组合截面的面积矩和形心位置的确定∑Aizcii=1nSy=∑Aiycii=1nSz=面积矩：yc=SzA∑Aiycii=1n=∑Aii=1nzc=SyA∑Aizcii=1n∑Aii=1n=形心位置：HOHAIUNIVERSITY5例2求图示截面的形心的位置。zy15050c5050C1C2C3解：21mm50150A22mm50180A23mm50250A250mm2551Cymm1402Cymm253Cy0321CCCzzzmm50250501805015025502501405018025550150321332211AAAyAyAyAyCCCCmm1200Cz15.5HOHAIUNIVERSITY6一、惯性矩的定义§A-2截面的惯性矩和惯性积Iy=z2dA∫AIz=y2dA∫A惯性矩恒为正二、惯性积的定义Iyz=yzdA∫AyzodAzydAdAyzzyz惯性积可正、可负或为零若y为对称轴，则Iyz=0HOHAIUNIVERSITY7三、形心主轴和形心主惯性轴主轴：惯性积为零的一对坐标轴。主惯性矩：截面对主轴的惯性矩。形心主轴：过截面形心的主轴。形心主惯性矩：截面对形心主轴的惯性矩。b/2b/2h/2h/2zyz'HOHAIUNIVERSITY8例3计算图示矩形对y轴和z轴的惯性矩和惯性积。b/2b/2h/2h/2zydyyb/2b/2h/2h/2zy解：AzAyId22/2/2hhbdyy123bh123hbIy0yzI同样地y、z为形心主轴Iy、Iz为形心主惯性矩HOHAIUNIVERSITY9例4计算图示圆形截面对其直径轴y和z的惯性矩。yzdzydyzyzd若为空心截面呢？（d/D）求Iy与Iz（作业题）464dIIzyHOHAIUNIVERSITY10四、惯性半径的定义iy=IyA√iz=IzA√故Iy=iyA2Iz=izA2注意平方问题第十六次课结束处HOHAIUNIVERSITY11§A-3惯性矩和惯性积的平行移轴公式一、平行移轴公式Iz=y2dA∫A=(a+yC)2dA∫A=a2dA∫A+2ayCdA∫A+yC2dA∫ACycayzcbzOycydAzczyCdA∫A对形心轴的面积矩=0yC2dA∫A对形心轴的惯性矩故Iz=a2dA∫A+IzC同理Iy=b2dA∫A+IyCIyz=abdA∫A+IyCzCHOHAIUNIVERSITY12二、组合截面惯性矩的计算式Iy=z2dA∫A=z2dA∫A1+z2dA∫An+…∑Iyii=1n=同理Iz∑Izii=1n=Iyz∑Iyzii=1n=HOHAIUNIVERSITY13例5图示矩形中，挖去两个直径为d的圆形，求余下图形对z轴的惯性矩。zyb/2b/243325121dbhIzHOHAIUNIVERSITY14例6由两个20a号槽钢截面图形组成的组合平面图形，设a=100mm,设求此组合平面图形对y,z两根对称轴的惯性矩。A=28.83×102mm2,Iyc=128×104mm4Izc=1780.4×104mm4,z0=20.1mmazyzCyCz0HOHAIUNIVERSITY15作业题求图示工字形截面对z轴的惯性矩。zbd