平截面假設和彈性力學精確解對比

1、關于材料力學平截面假設和彈性力學精確解的比較一、實驗目的：通過彈性力學逆解法求出簡單懸臂梁截面實際的位移的表達式和材料力學中利用橫截面假設計算的截面點進行比較；作出兩種方法求得截面實際圖形并比較；算出兩種方法中的誤差，驗證平截面假設的合理性；二、實驗步驟：確定梁的尺寸L,h,b；縱向施加的點載荷P的大??；確定材料屬性，幾何屬性并說明所取值對計算的合理性；逆解法推倒所得的橫截面位移表達式以及推倒方法；取x方向的五個截面并用材料力學的方法和彈性力學逆解法得到的公式分別得到5個截面的縱向圖形,并利用有限元軟件驗證彈性力學解和截面縱向圖形的正確性；以上圖形取21個點分析,并相對誤差,得出材料力學平截面

2、假設的合理性分析。三、實驗借助分析工具：1.MicrosoftExcel；2.AutoCad2007；3.MSC/Marc.四、實驗：模型（下圖）：用AutoCad2006建立模型的平面圖：該圖形為平面應力狀態(tài)下的懸臂梁,在懸臂梁上建立笛卡爾坐標系,以圖中梁的右端中點作為坐標原點。參數(shù)確定：為了保證梁的截面在載荷P作用下有比較明顯的彎曲變形，應該盡可能取大的載荷P,長度L值；使厚度和高度盡可能小。然而,為了保證梁受力彎曲后不至于發(fā)生塑性變形，應控制P值使梁的最大彎曲應力小于材料的屈服應力。材料參數(shù)確定：取該梁材料為Q460鋼，彈性模量E=200Gpa；泊松比近似取0.3；理論屈服強度為460M

3、pa幾何參數(shù)確定：如上圖，長度L=200m；高度c=10m；z向厚度取為1m；2.3利用Von.Mises屈服準則估算梁在純彎曲狀態(tài)下是否最大允許的載荷：僅受到彎曲應力時，等效應力Q二故由：QQ)2+QQ)2+QQ)22122331二Q彎曲彎曲MAXMyPLyMAX=IIzZ得PMA=76666666Pa，此處取P=60000000Pa分析。2.4彈性力學逆解法：利用位移分量求解得到橫截面位移u，v關于x，y的函數(shù)：Px2y2ETvPy36EI+空+(竺上6GT2EI2GIvPxy2Px3v二-+3.2位移計算結果:6EIPl2x2EIPl33EI在y方向上我選取20個點分析，發(fā)現(xiàn)u是關于y的

4、奇函數(shù)，v是關于y的偶函數(shù)，因此，在此只分析中性面以下（y軸正向）的11個點作比較分析（包括梁的中性面和截面的交點y=0）。注：點的標號設定：中性面點設為0,中性面往下依次設定為1，2，10；分別對應y=0；y=1；y=2；y=10.3.分析過程：（注：u方向對應x坐標方向；v方向對應y坐標方向）3.1選取x方向的5個截面分析：x=0；x=c；x=2c；x=0.51；x=l-c。利用Excel處理五個截面的數(shù)據(jù)，得到各截面的u，v向位移：彈性力學表達式可見，u是關于y的奇函數(shù)，v是關于y的偶函數(shù)；故此處先取y0的10個節(jié)點（點號1-10）分析，通過Excel軟件計算得出下表：（a）u方向的位移

5、：表3-1截面點號x=0 x=10 x=20 x=200 x=18000.000000E+00O.OOOOOOE+OOO.OOOOOOE+OOO.OOOOOOE+OOO.OOOOOOE+OO1-8.977635E-03-8.887635E-03-8.617635E-03-6.727635E-03-1.687635E-032-1.795541E-02-1.777541E-02-1.723541E-02-1.345541E-02-3.375410E-033-2.693346E-02-2.666346E-02-2.585346E-02-2.018346E-02-5.063464E-034-3.591

6、194E-02-3.555194E-02-3.447194E-02-2.691194E-02-6.751936E-0354.489097E02-4.444097E-02-4.309097E-02-3.364097E-02-8.440966E-036-5.387069E-02-5.333069E-025.171069E02-4.037069E-02-1.013069E-027-6.285126E-02-6.222126E-02-6.033126E-02-4.710126E-02-1.182126E-028-7.183280E-02-7.111280E-02-6.895280E-02-5.3832

7、80E-02-1.351280E-029-8.081545E-02-8.000545E-02-7.757545E-02-6.056545E-02-1.520545E-0210-8.979937E-02-8.889937E-02-8.619937E-02-6.729937E-02-1.689937E-02b）v方向位移：表3-2截面點號x=0 x=10 x=20 x=100 x=1801-1.200000E+00-1.020600E+00-8.448000E-01-3.750000E-01-1.740000E-022-1.200000E+00-1.020601E+00-8.448027E-01-

8、3.750068E-01-1.741215E-023-1.200000E+00T.020605E+00-8.448108E-01-3.750270E-01-1.744860E-024-1.200000E+00-1.020612E+00-8.448243E-01-3.750608E-01-1.750935E-025-1.200000E+00-1.020622E+00-8.448432E-01-3.751080E-01-1.759440E-026-1.200000E+00T.020634E+00-8.448675E-01-3.751688E-01-1.770375E-027-1.200000E+0

9、0T.020649E+00-8.448972E-01-3.752430E-01-1.783740E-028-1.200000E+00-1.020666E+00-8.449323E-01-3.753308E-01-1.799535E-029-1.200000E+00-1.020686E+00-8.449728E-01-3.754320E-01-1.817760E-0210-1.200000E+00-1.020709E+00-8.450187E-01-3.755468E-01-1.838415E-0211-1.200000E+00-1.020735E+00-8.450700E-01-3.75675

10、0E-01-1.861500E-02P（Lx）23.3通過材料力學的平截面假設分析u、v方向變形（同樣根據(jù)所f=一（引一4X）建坐標系只考慮y0部分）：y6EI3.1.1平截面假設：梁變形后額橫截面仍然保持平面且與變形后的梁軸線正交；計算得到；值得注意的是，上式計算所得撓度值得為負數(shù)，但目前坐標3.1.2計算公式：系設定為y在撓度增加的方向為正，計算個各截面v方向變形后（a）v向實際為梁各個截面的撓度值，這里可以用：坐標應該對撓度作適當?shù)恼撎柼幚怼?b)u向通過平截面假設作近似處理：首先在目前坐標系下應用轉角的表達式:P(L-X)29=y2EI8u=9fyy得到u方向的變形(下表)：表3-3

11、3.5應用有限元軟件MSC./Marc建模驗證以上計算的正確性：由上面縮確定的該梁的幾何性質可按照平面應力狀態(tài)對該梁作近似分析得到以下兩圖。圖中梁發(fā)生向下的微小彎曲變形，驗證了截面點號x=0 x=10 x=20 x=100 x=18000.000000E+00O.OOOOOOE+OOO.OOOOOOE+OOO.OOOOOOE+OOO.OOOOOOE+OO19.000239E-038.122675E-037.290125E-032.25OOO3E-O39.OOOOOOE-O521.800048E-021.624535E-021.458O25E-O24.5OOOO6E-O31.8OOOOOE-O4

12、32.700072E-022.436802E-022.187O38E-O26.75OOO8E-O32.7OOOOOE-O443.600096E-023.249070E-022.916O5OE-O29.OOOO11E-O33.6OOOOOE-O454.500120E-024.061337E-023.645O63E-O21.125OO1E-O24.5OOOOOE-O465.400143E-024.873605E-024.374O75E-O21.35OOO2E-O25.4OOOOOE-O476.300167E-025.685872E-025.1O3O88E-O21.575OO2E-O26.3OOOO

13、OE-O487.200191E-026.498140E-025.8321OOE-O21.8OOOO2E-O27.2OOOOOE-O498.100215E-027.310407E-026.561113E-O22.O25OO3E-O28.1OOOOOE-O4109.000239E-028.122675E-027.29O125E-O22.25OOO3E-O29.OOOOOOE-O4上面2.1-2.3中所取參數(shù)的可行性。Inc:0The.O.OOQe+OOO3335e-OD4-11?6b-0D1-2.4D2a-ODl-.fiDBe-OD：L-4A13E-0D1-34310-OD1-S.Ote-OOl1

14、Oe4e+ODO-12麗B+ODQDisplaceknitYi）圖3.1圖3.1左邊的標尺表明梁在x方向最大的變形為-0.0089m（和Excel圖表中計算值-0.0087相近），發(fā)生在梁的最右下端的附近點；標尺上數(shù)量級和Excel圖標中計算值一致；圖3.2（ii）圖3.2左邊的標尺表明梁在y方向最大的變形為-1.205m（和Excel圖標中計算值-1.200相近），發(fā)生在梁的最右方截面；標尺上數(shù)量級和計算值一致彈性力學方法和材料力學方法在梁變形后的對比分析：4.1按材料力學方法計算得在五個梁的截面y方向上不同點的坐標值:表4-1按表3-3中位移和各項原坐標疊加的結果：、截面方向坐標u1v1u

15、2v2u3v3u4v4u5v5-10-8.975437E-02-8.8000009.911146E+00-8.9792651.991385E+01-9.1549309.993275E+01-9.6243251.799831E+02-9.981385-9-8.078265E-02-7.8000009.920027E+00-7.9792911.992246E+01-8.1549819.993947E+01-8.6244531.799848E+02-8.981616-8-7.180976E-02-6.8000009.928910E+00-6.9793141.993107E+01-7.1550279.

16、994619E+01-7.6245681.799865E+02-7.981822-7-6.283582E-02-5.8000009.937794E+00-5.9793341.993968E+01-6.1550689.995291E+01-6.6246691.799882E+02-6.982005-6-5.386097E-02-4.8000009.946679E+00-4.9793511.994830E+01-5.1551039.995964E+01-5.6247571.799899E+02-5.982163-5-4.488534E-02-3.8000009.955565E+00-3.97936

17、61.995691E+01-4.1551339.996636E+01-4.6248311.799916E+02-4.982296-4-3.590906E-02-2.8000009.964451E+00-2.9793781.996553E+01-3.1551579.997309E+01-3.6248921.799933E+02-3.982406-3-2.693225E-02-1.8000009.973338E+00-1.9793881.997415E+01-2.1551769.997982E+01-2.6249391.799949E+02-2.982491-2-0.017955-0.800000

18、9.982225E+00-0.9793951.998276E+01-1.1551899.998654E+01-1.6249731.799966E+02-1.982551-1-8.977590E-030.2000009.991112E+000.0206011.999138E+01-0.1551979.999327E+01-0.6249931.799983E+02-0.98258800.000000E+001.2000001.000000E+011.0206002.000000E+010.8448001.000000E+020.3750001.800000E+020.01740018.977590

19、E-032.2000001.000889E+012.0206012.000862E+011.8448031.000067E+021.3750071.800017E+021.01741221.795505E-023.2000001.001778E+013.0206052.001724E+012.8448111.000135E+022.3750271.800034E+022.01744932.693225E-024.2000001.002666E+014.0206122.002585E+013.8448241.000202E+023.3750611.800051E+023.01750943.590

20、906E-025.2000001.003555E+015.0206222.003447E+014.8448431.000269E+024.3751081.800067E+024.01759454.488534E-026.2000001.004444E+016.0206342.004309E+015.8448681.000336E+025.3751691.800084E+025.01770465.386097E-027.2000001.005332E+017.0206492.005170E+016.8448971.000404E+026.3752431.800101E+026.01783776.

21、283582E-028.2000001.006221E+018.0206662.006032E+017.8449321.000471E+027.3753311.800118E+027.01799587.180976E-029.2000001.007109E+019.0206862.006893E+018.8449731.000538E+028.3754321.800135E+028.01817898.078265E-0210.2000001.007997E+0110.0207092.007754E+019.8450191.000605E+029.3755471.800152E+029.0183

22、84108.975437E-0211.2000001.008885E+0111.0207352.008615E+0110.8450701.000673E+0210.3756751.800169E+0210.0186154.2按彈性力學方法計算得在五個梁的截面y方向上不同點的坐標值：表4-2按表3-1及3-2中位移和各項原坐標疊加的結果截面坐標Y方向坐標u1v1u2v2u3-10-9.000239E-02-8.8000019.918773E+00-8.889926-9-8.100215E-02-7.8000019.926896E+00-7.889926-8-7.200191E-02-6.8000

23、019.935019E+00-6.889926-7-6.300167E-02-5.8000019.943141E+00-5.889926-6-5.400143E-02-4.8000019.951264E+00-4.889926-5-4.500120E-02-3.8000019.959387E+00-3.889926-4-3.600096E-02-2.8000019.967509E+00-2.889926-3-2.700072E-02-1.8000019.975632E+00-1.889926-2-1.800048E-02-0.8000019.983755E+00-0.889926-1-9.00

24、0239E-030.1999999.991877E+000.1100740O.OOOOOOE+OO1.1999991.000000E+011.11007419.000239E-032.1999991.000812E+012.11007421.800048E-023.1999991.001625E+013.11007432.700072E-024.1999991.002437E+014.11007443.600096E-025.1999991.003249E+015.11007454.500120E-026.1999991.004061E+016.11007465.400143E-027.199

25、9991.004874E+017.11007476.300167E-028.1999991.005686E+018.11007487.200191E-029.1999991.006498E+019.11007498.100215E-0210.1999991.007310E+0110.110074109.000239E-0211.1999991.008123E+0111.110074v3u4v4u5v51.992710E+01-8.9794019.997750E+01-9.6250001.799991E+02-9.9826001.993439E+01-7.9794019.997975E+01-8

26、.6250001.799992E+02-8.9826001.994168E+01-6.9794019.998200E+01-7.6250001.799993E+02-7.9826001.994897E+01-5.9794019.998425E+01-6.6250001.799994E+02-6.9826001.995626E+01-4.9794019.998650E+01-5.6250001.799995E+02-5.9826001.996355E+01-3.9794019.998875E+01-4.6250001.799995E+02-4.9826001.997084E+01-2.97940

27、19.999100E+01-3.6250001.799996E+02-3.9826001.997813E+01-1.9794019.999325E+01-2.6250001.799997E+02-2.9826001.998542E+01-0.9794019.999550E+01-1.6250001.799998E+02-1.9826001.999271E+010.0205999.999775E+01-0.6250001.799999E+02-0.9826002.000000E+011.0205991.000000E+020.3750001.800000E+020.0174002.000729E

28、+012.0205991.000023E+021.3750001.800001E+021.0174002.001458E+013.0205991.000045E+022.3750001.800002E+022.0174002.002187E+014.0205991.000068E+023.3750001.800003E+023.0174002.002916E+015.0205991.000090E+024.3750001.800004E+024.0174002.003645E+016.0205991.000113E+025.3750001.800005E+025.0174002.004374E

29、+017.0205991.000135E+026.3750001.800005E+026.0174002.005103E+018.0205991.000158E+027.3750001.800006E+027.0174002.005832E+019.0205991.000180E+028.3750001.800007E+028.0174002.006561E+0110.0205991.000203E+029.3750001.800008E+029.0174002.007290E+0111.0205991.000225E+0210.3750001.800009E+0210.0174004.3兩種

30、方法所得的坐標圖比較：（注：圖中系列1代表材料力學計算方法：系列二代表彈性力學計算方法）圖4. (2)X=10截面(1)x=o截面x=o截面-10.000000-1.(D0000-8.()()()()()-6.()()()()()-4.()()()()()()()()()()().0(0E-01)000001系列圖4. (3)X=20截面X=10u圖4. (4)X=100X=20圖4. (5)X=180X=100圖4. X二18015.00000010.0000005.000000V0.0000001.7S系列0200E029800E1.799850E1.799900E1.7999.S0E1.

31、8|0000El.800050E1.800100E1.800150E1.8()2+02+02用Ht02+02+02+02+(-5.000000-10.000000-15.000000(5)彈性力學方法算得的各截面在坐標上的反映:u彈性力學各截面U圖4.6(6)材料力學方法算得的各截面在坐標上的反映材料力學各截面15.000000ii10.000000：i5.0000001i|5：1Iiiiiiiri0.000000“-2.00000.0(00E+01OEiboo002.000004.000006.000008.000001.000001.200001.400001.600001.8(0E+010E+010E+010E+01Oe4o20E+020E+020E+020E(0002.00000020E+02-5.000000iii：V-10.000000X=10X=20X=100-15.000000兩種方法計算結果的比較：先把5個截面u,v方向上用兩種方法計算所得位移的差值疊加后取平均值(因為兩種方法計算得的u是關于y的奇函數(shù)，所以只需取從0點開始的朝y正向的11個點；v方向則計算所有點)，得到u方向位移差值的數(shù)學期望E；再計算這些點的方差。5.1數(shù)學期望E：截面XIX2X3