截面的幾何性質(zhì)

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1、附錄附錄ⅠⅠ截面的幾何性質(zhì)截面的幾何性質(zhì)?xdA一、截面的形心和靜矩Ax?形心cA(centroid)O?ydAyAyxyc?CA靜矩靜矩(staticmoment)yCCdAAS??ydAyAxc?zASx??dAxAyc?Ax??同一截面靜矩可正，同一截面靜矩可正，可負也可為零?？韶撘部蔀榱?。?圖形對通過形心的軸的靜矩恒為零圖形對通過形心的軸的靜矩恒為零;;反之，若反之，若圖形對某軸的靜矩為零，則該軸必通過圖形的形圖形對某軸的靜矩為零，則該軸必通過圖形的形心，并稱該軸為心，并稱該軸為形心軸形

2、心軸。。Thestaticmomentoftheareawithrespecttoitscentroidaxisiszero.?組合圖形的靜矩和形心Sy??ddAyASx??AxAxc??ycAA靜矩靜矩nnnSx??Sxi??Aiyci;Sy??Aixci1i?1i?1100ynnO形心??AxAyⅠC20SiciSici1xy??yix??11??iccAAAAC例1：計算組合圖形的形心C140SyAx11cc?Ax222x??cⅡAA?A12100201014020????????2070

3、?mmx1002014020???20=56.6mm二、慣性矩(Momentofinertia)和慣性積(Productofinertia)2Oy極慣性矩：極慣性矩：IAp???dyxA?Cy2CCI?yAddAx?A慣性矩：慣性矩：Az2I?xAdy?Ax慣性積：I?xyAdxy?AIx2222i?,I?AiIA????d()xydA慣性慣性xAxxp??AA半徑：半徑：I??IIy2xyi?,I?AiyyyA例2：計算矩形對x軸和y軸的慣性矩。2b/2b/2I?yAdx?A慣性矩：慣性矩：2

4、I?xAdy?Ah/2Oxy3b/2hbh/2dxIy??2hdyx??b/2123dyybhxIy?12例例33：：計算圓形對其直徑軸計算圓形對其直徑軸xx和和yy的慣性矩。設(shè)的慣性矩。設(shè)圓的直徑為圓的直徑為dd。。d4?dI?I?I?pxy324y?dφxIx?Iy?dx64x三、平行移軸公式(Parallelaxistheorem)Oy慣性矩：慣性矩：22Iy??ddAIxAxy??bxAACxxxc、、yyc為為一對形心軸，一對形心軸，cycx∥xydAc，y∥yc。。則：則：acyIy

5、???22d()AyaAdxxcxC??AA22??yd2AaydAa?dA?AAACC??2??IaAxc2同理：Iy?Iyc?bAIxy?Iaxycc?bA例4：求圖示T字形截面的形心主慣性矩。1001.確定形心yOxy??56.6mm，50mm20CC2.求IIxc，ycCyc1133I??????2010014020xC1401212644??1.7610mm?176cm132I??????1002010020(56.610)?xyC2012xc132???????2014014020(

6、9056.6)12744??1.21110mm?1211cm慣性積I?xyAd慣性積：xy?AOyyx?C?同一截面慣性積可正，可負也可為零。yCCdAAz?若兩根正交軸中有一根坐標軸是截面的對稱軸，則圖形對這對軸的慣性積必為零。x?慣性積為零的一對軸，稱為平面圖形的主慣性軸（主軸）,平面圖形對主軸的慣性矩稱為主慣性矩。?當(dāng)主軸通過截面形心時，xc、ycc——形心主軸,Ixc、Iycyc——形心主慣性矩。四、轉(zhuǎn)軸公式(Formulaofrotationofaxes)xO???DOGGDy1y1?

7、?OGFEHAx1C???OFcos?AFsin?DBxycossinG?????ExOFyA???DAEDE1??AEGF坐標轉(zhuǎn)換的矩陣形式??AFcos?OFsin???x1?cos??sin???x?????????yxcos??sin??y??sin??cos???y1已知：截面對y、z軸的慣性矩、慣性積Iy,Iz,Iyz求解：截面對y1、z1軸的慣性矩、慣性積Iy1,Iz1,Iy1z122Iy???d(AyxAcoss??in)dx1??12222??????yAcos?dxAsind

8、??2xysincosd?A1cos2????221cos2????yAdd??xAsin2?xyAd221cos2????1cos2???III(sin2)?xyxy22IIII??xyxy??cos2??Isin2?xy22IIII??xyxyII??cos2??sin2?xx1y22IIII??xyxyII??cos2??sin2?yx1y22II?xyII??sin2?cos2?xy11xy2顯然IIIII????x11yxyp2Ixy由方程tg2???求解出?1,?2II?xy12Ix