三角形悬臂梁应力分析三角形悬臂梁应力分析摘要：在有限元分析软件ANSYS12.0平台上建立三角形悬臂梁的力学模型，添加约束和载荷，计算出应力分布，并与理论计算值相比较。

⒈ 引言目前，ANSYS 软件具有其强大的功能已经被广泛的应用于机械，化工，土木，交通等各个领域。

应用ANSYS 分析，可以大大减少人力物力的投入，而且可靠性高，对于三角形悬臂梁分析其应力和变形情况，分析方法和结论可作为这类设计的参考。

⒉ 计算模型Ⅰ问题描述【三角形悬臂梁忽略重力作用，∠BAC=α，AB 边上作用均布载荷q ，求应力的解析表达，计算出BC 边上的应力值并与ANSYS 计算值比较，绘出应力曲线图】选取应力函数：Ansys 计算参数值：AB=1000mm ，α=30°，厚度t=20mm2222[()sin cos cos tan ]C r r r ϕθθθθα=∂-+-Ⅱ解析解根据弹塑性平面问题的极坐标解答，利用以下公式推导：222222211111()r r r r r rr r r r r θθϕϕσθϕσϕϕϕτθθθ∂∂=+∂∂∂=∂∂∂∂∂=-=-∂∂∂∂∂ 以及2222cos sin 2sin cos sin cos 2sin cos x r r y r r θθθθσσθσθτθθσσθσθτθθ=+-=++已知2222[()sin cos cos tan ]C r r r ϕθθθθα=∂-+-， 故有以下式子成立：22222222222[2()2sin cos 2cos tan ][2()2sin cos 2cos tan ][cos 2sin 2tan ][2sin 22cos 2tan ]C r r r rC r C r r r C r r ϕαθθθθαϕαθθθθαϕθθαθϕθθαθ∂=-+-∂∂=-+-∂∂=-++∂∂=-+∂ 所以，22222222211[2()sin 22cos tan 2cos 2tan ][2()sin 22cos tan ]111()[1cos 2sin 2tan ]r r C r r r C rC r r r r r θθϕϕσαθθθαθαθϕσαθθθαϕϕϕτθθαθθθ∂∂=+=---+∂∂∂==-+-∂∂∂∂∂=-=-=--∂∂∂∂∂ 因此，222222222224cos sin 2sin cos [2()2sin 2cos 2cos tan 2cos cos 2tan sin 2cos 2sin 2tan ]sin cos 2sin cos [2()2cos sin 2cos 2sin 2tan 2tan sin cos 3tan cos ]x r r y r r C C θθθθσσθσθτθθαθθθθαθθαθθθασσθσθτθθαθθθθθααθθαθ=+-=---+++=++=-+-++- 由边界0()/y y q t σ==-,即当0θ=时，/y q t σ=-；带入y σ的表达式中可得：2(tan )q C t αα=- 将C 带入x σ表达式中得：22222222[2()2sin 2cos 2cos tan 2cos cos 2tan sin 2cos 2sin 2tan ][2()2sin 2cos 2cos tan 2(tan )2cos cos 2tan sin 2cos 2sin 2tan ]x C q t σαθθθθαθθαθθθααθθθθαααθθαθθθα=---+++=----+++Ⅲ Ansys 计算问题详细说明：取AB=1000mm ，30α=,厚度t=20mm ，50/q N mm =-。

解题过程：① 前处理，设置材料属性，E=200e3，泊松比0.3μ=，单元类型为solid 20 node 95；建立几何模型，有点生成面，再由面生成体；单元网格划分为三角形。

② 加载求解，加约束，面约束，加载荷，均布载荷50/q N mm =-，并求解。

③ 后处理⒊ 结果分析变形图计算云图等效应力曲线与理论值相比较，利用ANSYS计算的结果很接近于理论值，采用有限元法对悬臂梁进行应力分析具有很高的效率和可靠性。

参考文献S.P. Timoshenko J.N.Goodier，Theory of Elasticity,北京，清华大学出版社,2004.刘相新，孟宪颐，ANSYS基础与应用教程，北京，科学出版社，2006. 王金龙，王清明，王伟章，ANSYS12.0有限元分析与范例解析，机械工业出版社，2010.LOG文件：/BATCH/COM,ANSYS RELEASE 12.0.1 UP20090415 11:33:07 01/11/2012!*/NOPR/PMETH,OFF,0KEYW,PR_SET,1KEYW,PR_STRUC,1KEYW,PR_THERM,0KEYW,PR_FLUID,0KEYW,PR_ELMAG,0KEYW,MAGNOD,0KEYW,MAGEDG,0KEYW,MAGHFE,0KEYW,MAGELC,0KEYW,PR_MULTI,0KEYW,PR_CFD,0/GO!*/COM,/COM,Preferences for GUI filtering have been set to display: /COM, Structural!*/PREP7!*ET,1,SOLID95!*!*MPTEMP,,,,,,,,MPTEMP,1,0MPDATA,EX,1,,200e3MPDATA,PRXY,1,,0.3SAVEK,1,0,577,0,K,2,1000,577,0,K,3,1000,0,0,FLST,2,3,3FITEM,2,1FITEM,2,2FITEM,2,3A,P51X!*VOFFST,1,20, ,/VIEW,1,1,1,1/ANG,1/REP,FASTSAVE/DIST,1,1.08222638492,1 /REP,FAST/DIST,1,1.08222638492,1 /REP,FAST/DIST,1,1.08222638492,1 /REP,FAST/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,1.08222638492,1 /REP,FAST/DIST,1,1.08222638492,1 /REP,FAST/DIST,1,1.08222638492,1 /REP,FAST/DIST,1,1.08222638492,1 /REP,FAST/DIST,1,1.08222638492,1 /REP,FAST/DIST,1,1.08222638492,1 /REP,FAST/DIST,1,1.08222638492,1 /REP,FASTSAVEFLST,2,1,5,ORDE,1 FITEM,2,4!*DA,P51X,ALL,FLST,2,1,5,ORDE,1 FITEM,2,3/GO!*SFA,P51X,1,PRES,50 SAVEESIZE,0,20, MSHAPE,1,3D MSHKEY,0!*CM,_Y,VOLU VSEL, , , , 1 CM,_Y1,VOLU CHKMSH,'VOLU' CMSEL,S,_Y!*VMESH,_Y1!*CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2!*FINISH/SOL/STATUS,SOLU SOLVE/VIEW,1,,,1/ANG,1/REP,FAST/VIEW,1,1,1,1/ANG,1/REP,FAST/VIEW,1,,,1/ANG,1/REP,FASTFINISH/POST1INRES,ALLFILE,!*PLDI,1,ANMODE,10,1, ,0/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,1.08222638492,1 /REP,FAST/DIST,1,1.08222638492,1 /REP,FAST/DIST,1,1.08222638492,1 /REP,FAST/DIST,1,1.08222638492,1 /REP,FAST/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,1.08222638492,1 /REP,FASTSET,LISTPLDISP,1PLDISP,2PLDISP,1PLDISP,2!*/EFACET,1PLNSOL, S,EQV, 0,1.0!*PLESOL, S,EQV, 0,1.0 SET,LIST,2PRITERPRERR!*NSORT,U,X,0,0, ,0!*!*NSORT,U,X,0,0, ,0!*NSORT,U,Y,0,0, ,0!*!*NSORT,U,Z,0,0, ,0!*!*!*PRNSOL,S,COMP!*PRNSOL,S,PRIN!*AVPRIN,0,0,ETABLE, ,S,X!*!*PRETAB,SX!*PLETAB,SX,AVG/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,0.924021086472,1 /REP,FAST/DIST,1,1.08222638492,1 /REP,FAST/DIST,1,1.08222638492,1 /REP,FAST/DIST,1,1.08222638492,1 /REP,FAST/DIST,1,1.08222638492,1 /REP,FASTSET,LIST,2PLDISP,1PLDISP,1FLST,2,42,1FITEM,2,2FITEM,2,1799FITEM,2,46FITEM,2,48FITEM,2,50FITEM,2,52FITEM,2,528FITEM,2,54 FITEM,2,55 FITEM,2,56 FITEM,2,58 FITEM,2,60 FITEM,2,61 FITEM,2,62 FITEM,2,63 FITEM,2,64 FITEM,2,66 FITEM,2,543 FITEM,2,67 FITEM,2,68 FITEM,2,70 FITEM,2,71 FITEM,2,73 FITEM,2,74 FITEM,2,75 FITEM,2,76 FITEM,2,78 FITEM,2,1759 FITEM,2,1962 FITEM,2,1713 FITEM,2,542 FITEM,2,527 FITEM,2,526 FITEM,2,51 FITEM,2,53 FITEM,2,57 FITEM,2,59 FITEM,2,69 FITEM,2,77 FITEM,2,1829 FITEM,2,45 FITEM,2,49!*PATH,1,42,30,20, PPATH,P51X,1 PATH,STAT!*/PBC,PATH,1/REPLOT/PBC,PATH,0!*PLSECT,S,EQV, ,1/DIST,1,1.08222638492,1 /REP,FASTPRSECT,1,1PRSECT,1,1PCALC,ADD, , , ,1,1, , PCALC,ADD, , , ,1,1, , PCALC,EXP,1, , ,1,0,/DIST,1,0.924021086472,1 /REP,FASTFINISH! /EXIT,MODEL。