常用截面几何特性计算公式

　第1章常用资料、数据和一般标准G1常用几何体的体积、面积及重心位置�表G1-1�表G1-1常用几何体的体积、面积及重心位置图形体积V、底面积A、侧面积A0、全面积An、重心位置G的计算公式图形体积V、底面积A、侧面积A0、全面积An、重心位置G的计算公式2)(364G2n2023aZdadaAaAaAaV======为对角线)3(4)4()2(p)2(p)(pp2p)3(3p)3(6pG222n2202222hrhrhZaharhAharhAaAhrhhahV−−=+=+=+===−=+=2)()(2)(2G222n0hZdhbadbhahabAbahAabAabhV=++=++=+===为对角线在椭球中心重心GabcVp34=与球心重合重心GrArV2n3p4p34==在圆环中心重心GDdRrADdRrV22n2222pp44pp2====rZrArArArV83p3p2pp32G2n2023=====V=πr2hA0=2πrhAn=2πr(r+h)ZG=2h2(续)图形体积V、底面积A、侧面积A0、全面积An、重心位置G的计算公式图形体积V、底面积A、侧面积A0、全面积An、重心位置G的计算公式2))((p2)(p2)(p)(pGn02222hZhrRrRArRhArRArRhV=+−+=+=−=−=)22(2)3(])(4)()(4)[(21)22(611111111G01n2121212101111111babaababbabaababhZAAAAbbhaaaahbbAabAbaAbabaababhV++++++=++=−+++−++===+++=)(4)32()()(p)(p)(3p2222G22022n022rRrRrRrRhZhrRlArRArRlARrrRhV++++=+−=++=+=++=44232332331G0n22022hZAAAalaAaAhaAhV=+=−====4)(pppp31G22n022hZhrllrrArlArAhrV=+=+====)()(4)32()(3233233131212212G01n10221211111为斜高为顶面积,gAaaaaaaaahZAAAAaagAaAaAaaaahAV++++=++=+===����������������++=4)44(21)44(2131G2222n22220hZbhaahbabAbhaahbAabAabhV=++++=+++===2)(46336233233G222n022hZdahdahaAahAaAhaV=+=+====为对角线3G2常用力学公式G2.1常用截面的力学特性(表G1-2、表G1-3)表G1-2常用截面的几何及力学特性截面形状面积A惯性矩I截面系数eIW=回转半径AIi=形心距离e2a124a3x12x1179.06aWaW==aa289.012=aeae7071.02x1x==22ba−1244ba−abaWabaW44x144x1179.06−=−=22289.0ba+aeae7071.02x1x==ab123ab62abbb289.012=2b)(hHb−12)(12)(3y33xhHbIhHbI−=−=6)(6)(2y33xhHbWHhHbW−=−=bihHhHi289.012y22x=++=22yxbeHe==)(2baH+32)(3642Hbaabba+++)2(12)4()2(12)4(222xb222xabababaHWbababaHW+++=+++=24)(322bababaH++×+)(3)2(babaH++2bH363bH12242xb2xabHWbHW==HH236.023=3H4(续)截面形状面积A惯性矩I截面系数eIW=回转半径AIi=形心距离eRCCA==2598.2xyxIIRI==45413.03y3x5413.0625.0RWRW==Ri4566.0x=ReRe==yx866.04p2d64p4d32p3d4d2d)(4p22dD−)(64p44dD−��������−DdD4432p444dD+2D4p22da−��������−16p312144da��������−16p36144daa)p4(48p3162244dada−−2a8p2d128p00686.04y4xdIdI==64p0239.03y4xdWdW==41319.0yxdidi==dyde2122.02878.0sx==8)(p22dD−128)(p)(00686.044y44xdDIdDI−=−=��������−=443y164pDddW22yx41dDFIiAIiyx+===)(p3)(222sdDdDdDy+++=sxx24y2s1xx431x22222sinsin32sin1808cossin8842184296.57)2(201745.0)]([21yrJWrIAyJJrlrIcrrhhhcrrlhrhclhrcrlA−=������−−°=−=−=−−=+==−==−−=�����ααααπααααAIixx=Acy123s=5(续)截面形状面积A惯性矩I截面系数eIW=回转半径AIi=形心距离eabp4p4p3y3xbaIabI==4p4p2y2xbaWabW==22yxaibi==aebe==yx)(p11baab−)(4p)(4p1313y3113xbabaIbaabI−=−=ababaWbbaabW4)p(4)(p1313y3113x−=−=AIiAIiyyxx==aebe==yx)(2hebBH+−333231xbhaeBeI−+=2x2x1x1xeIWeIW==)]([3233231hebHBbhaeBe+−−+12221)(2eHebtaHbtaHe−=++=6(续)截面形状面积A惯性矩I截面系数eIW=回转半径AIi=形心距离ebhBH+1233xbhBHI+=HbhBHW633x+=)(1233bhBHbhBH++2HbhBH−1233xbhBHI−=HbhBHW633x−=)(1233xbhBHbhBHi−−=2H7表G1-3主要组合截面的回转半径截面形状回转半径截面形状回转半径hihiYX215.030.0==bihiYX21.021.0==bihiYX20.032.0==bihiYX43.043.0==bihiYX24.028.0==bihiYX22.042.0==bihiYX17.030.0==bihiYX20.039.0==bihiYX21.026.0==bihiYX56.035.0==hibihiZYX185.021.021.0===bihiYX60.038.0==8(续)截面形状回转半径截面形状回转半径bihiYX44.038.0==bihiYX24.045.0==235.0dDddicpcpX+==bihiYX21.040.0==bihiYX38.044.0==bihiYX235.045.0==bihiYX54.037.0==bihiYX32.044.0==bihiYX45.037.0==9G2.2受静载荷梁的支点反力、弯矩和变形计算公式�表G1-4、表G1-5�表G1-4常用静定梁的支点反力、弯矩和变形计算公式序号载荷情况及剪力图弯矩图支点反力弯矩方程挠度曲线方程最大挠度梁端转角12FFFBA==2)(:2/≤≤0FxxMlx=��������−−=3334348:2/≤≤0lxlxEIFlylxEIFlylx48:2/3max−==处在EIFl162BA−=−=θθ2lFaFlFbFBA==)(1)(:1≤≤1)(:≤≤0axFFbxxMxaFbxxMax−−==��������−+−−×−=−−−=baxxxblEIlFbylxbxlEIlFbxyax332222)()(6:≤≤0)(6:≤≤02EIblFbylxEIlblFbyblxba48)43(:2/39)(:3,222/322max22−−==−−=−=处在处在若EIlalFabEIlblFab6)(6)(BA+=+−=θθ3FFFBA==FaMalxaFxxMax=−=:≤≤)(:≤≤0])(3[6:≤≤0])(3[6:≤≤022axlxEIFayalxxalaEIFxylx−−−=−−−−=)43(24:2/22maxalEIFaylx−−==处在)(22BAalEIFa−−=−=θθ910(续)序号载荷情况及剪力图弯矩图支点反力弯矩方程挠度曲线方程最大挠度梁端转角4lMFFBA==������−=lxMxM1)(��������+−−=33222326lxlxlxEIMlyEIMlylxEIMlylx16:2/39:31122max−==−=��������−=处在处在EIMlEIMl63BA=−=θθ5lMFFBA==lMxxM=)(��������−−=3326lxlxEIMlyEIMlylxEIMlyx16:2/39:3122max−==−==处在处在EIMlEIMl36BA=−=θθ6lMFFBA==������−=−=lxMxMlxalMxxMax1)(:≤≤)(:≤≤0])(3[6)(:≤≤)3(6:≤≤0222222xlalEIlxlMylxaxblEIlMxyax−−−−−=−−=EIlalMyalxEIlblMyblx39)3(:3/)3(39)3(:3/)3(2/322max2222/322max122−−=−=−=−=处在处在)33(66)3(6)3(222C22B22AlbaEIlMEIlalMEIlblM−+−=−=−=θθθ1011(续)序号载荷情况及剪力图弯矩图支点反力弯矩方程挠度曲线方程最大挠度梁端转角72qlFFBA==)(2)(xlqxxM−=)2(24323xlxlEIqxy+−−=EIqlylx3845:2/4max−==处在EIql243BA−=−=θθ83600lqFlqFBA==��������−=22016)(lxlxqxM��������+−×−=5533403107360lxlxlxEIlqyEIlqylx30max00652.0:519.0−==处在EIlqEIlq45360730B30A=−=θθ1112(续)序号载荷情况及剪力图弯矩图支点反力弯矩方程挠度曲线方程最大挠度梁端转角9������+=������+=cblqbFcblqbFBA22������������++×������+=������++=−−������+=+������+=cblbacblqbMcblbaxaxqxcblqbxMbaxaxcblqbxMax222:2)(22)(:≤≤2)(:≤≤0max2处在��������−−−��������+−×−+−=+���−+����−−�����������������+−×��������+−=+��������−−��������+−×��������+−=2222422222222)(42))((6≤≤)(44226:≤≤4226:≤≤0xlbbalxlbaEIlqbylxbaaxblxbcblxcbEIlqbybaxaxbcblcbEIlqbxyaxmax,,0=':+≤≤yyxybaxa方程即得代入的数值解求出令处在��������−������+−×+=��������−������+−×������+−=42)(64226222B222AbbalbaEIlqbbcblcbEIlqbθθ1040lqFFBA==��������−=2204312)(:2/≤≤0lxlxqxMlx��������+−−=553340164025960:2/≤≤0lxlxlxEIlqylxEIlqylx120:2/40max−==处在EIlq19250BA−=−=θθ1213(续)序号载荷情况及剪力图弯矩图支点反力弯矩方程挠度曲线方程最大挠度梁端转角11llaFFlFaFBA)(+==)()(:≤≤)(:≤≤0xalFxMalxllFaxxMlx−+−=+−=])()([6:≤≤6:≤≤0332332lxlaaxxal