材料力学附录 截面的几何性质-正式

1、123OyzdAyz AyAzSd AzAySd4yzO dA yzCzyASAAzzyA dASAAyyzA dzASy yASz 平面图形对某一轴的静矩等于图形面积乘以相应的形心坐标平面图形对某一轴的静矩等于图形面积乘以相应的形心坐标。形心又称面积中心，对于均质等厚的形心又称面积中心，对于均质等厚的薄板，形心也是薄板的重心。薄板，形心也是薄板的重心。5 6zASiniiy 1 niiizyAS1),(yzii niiniiiAzAz11 niiniiiAyAy117AAyAyAAyAyniiniii21221111 AAzAzAz212211 101012012Ozy901y1z2z2y8

2、211200mm12010 A5mm1 y60mm1 z22800mm8010 A50mm280102 ymm25 z38mm23mm212211212211 AAzAzAzAAyAyAy101012012Ozy901y1z2z2y9C2C1yz 212211AAAyAyAAyyii221108090120)11080(5120901010例例2 2 求图示半圆形的形心位置。求图示半圆形的形心位置。z由对称性，只需求由对称性，只需求 。zzzRzyAd2dd22302232d2dRzzRzAzSRAy34213223RRRASzy11yzOdAyz AzAyAyIAzIdd22AIiyy AI

3、izz 12yzOdAyz zyAAAApIIAyAzAzyAI dd dd22222zypIII ApAId213yzOdAyz AyzAyzIdyzdydyzdAdA14bhyzCzdzAzIAyd2 zbAdd12dd32222bhzbzAzIhhAy 123hbIz 15 zyd 32242/022PdrdrrdArIdA PIIIzy zyII 6424dIIIpzy 所以：所以：zyrdrR16例例5 5 求圆环的求圆环的yzII ,小大PPPIII44321321dD Dd小大zzzyIIIIdDyz)1(32144 D)1(64144 D17yzOC（b,a)ba18yzOC(

4、b,a)bazCyC 19yczcyczcOyzzy求求: :,yzyzIII已知已知 CzyyzCCCI ,I ,I解：解：azzC byyC dAbaCdAzIAy 2dAazAC 2)(dAadAazdAzAACAC 222 CyIyaS2Aa2 0AaICy2 20AaIICyy2 AbIICzz2 dAyzIAyzdAbyazACC)(dAabdAaydAbzdAzyAACACACC0abAICCzyyczcyczcOyzzydAbaC0azzC byyC 21平行移轴公式：平行移轴公式：AaIICyy2 AbIICzz2 abAIICCzyyz 注意：注意：22 niyiyII1

5、nizizII1 niyziyzII1yziziyiIII,23201401002021zC yC140201A8010701 z201002A0 z27mm.46212211 AAzAzAzCy1zcz242014010020y21zcyC2z46-1223I1069. 71014020)7 .4680(14020121mICy 122310)7 .46(2010020100121 】【IIyCI46m1012.12 IIyIyyCCCIIIAaIICyy2 46103.44m 7mm.46212211 AAzAzAzCcz25Oyzy1z1 26sincos1zyysincos1yzz解解

6、:oyzAdAyz1z1y1y1z27AydAzI211AdAyz2)sincos(AdAz22cosAyzdAcossin2AdAy22sin222sinIsinIcosIzyzy三角函数关系: 2cos121cos22cos121sin22sincossin2 2sin2cos221yzzyzyyIIIIII28AzydAzyI1111AdAyzzy)sincos)(sincos(AdAyzcossin)(22AyzdA)sin(cos22 2cos2sin)(21yzzyIII29IIIIIIyzzyzyy2sin2cos221 IIIIIIyzzyzyz2sin2cos221 IIII

7、yzzyzy2cos2sin211 zyzyIIII 1130 IIIIyzzyzy2cos2sin211 1112)2cos2sin2(2zyyzzyyIIIIddI 3102cos2sin20000 yzzyzyIIIIzyyzIIItg220220204)(2tan112cosyzzyzyIIIII 220004)(22cos2tan2sinyzzyyzIIII IIIIIIyzzyzyy2sin2cos221 IIIIIIyzzyzyz2sin2cos221 32224)(21200yzzyzyzyIIIIIII00minmax zyIIII 33 iyyII izzII iizyyz

8、II)2(tan2zyyzIII 10 224)(2200yzzyzyzyIIIIIII 1 niiniiiAyAy11 niiniiiAzAz1134101012025C4020yz 20158035AaIICyy2 AbIICzz2 abAIICCzyyz 2323257010121101201510120121）（yI44mm104.1001070 224)(2200yzzyzyzyIIIIIII 135mm104 .27844 zI093. 1)2(tan20 IzIyIyz zyII 02 6 .22720 8 .1130 mm103 .971070)35()25(0101202015044 yzI101012025C4020yz 2015803536101012070C4020yzy0 0=113.8z0mm104 .3214)(21244220 yzzyzyyIIIIIImm104 .574)(21244220 yzzyzyzIIIIII37ddddddAAzzAAAyyiiii177. 0425 . 14200022 CyICzICCzyIdb2dyzOyCzCC3842242232 2 685. 0)177. 05 . 0(464)177. 0(312)2(5 . 1)5 . 0( 1dddddddddzdAIzAIII