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2Theoryofplaneproblems2.1PlaneStressandPlaneStrain2.2DifferentialEquationsofEquilibrium2.3StressataPoint.PrincipalStresses2.4GeometricalEquations.Rigid-bodyDisplacements2.5StrainataPoint2.6PhysicalEquations2.7BoundaryConditions2.8Saint-Venant’sPrinciple2.9SolutionofPlaneProbleminTermsofDisplacements2.10SolutionofPlaneProbleminTermsofStresses
2.12Airy’sStressFunction.InverseMethodandSemi-InverseMethod2.11CaseofConstantBodyForces2.1.PLANESTRESSANDPLANESTRAINIfAbodyhasaparticularshape.
Theexternalforcesaredistributedinaparticularmanner.Thespatialproblem
PlanestressproblemPlanestrainproblemAplaneproblem
1)Planestress
a.Theshapeofthebody:Thinplate:thicknesstthelengthsoftheotherdimensionsxyozoytb.Mannerofexternalforces:Surfaceforces:loadontheedges,areparalleltothefacesand
distributeuniformlyoverthethickness.Bodyforces:areparalleltothefacesand
distributeuniformlyoverthethickness.xyozoytSincetheplateisverythinandtheexternalforcesareuniformlydistributedoverthethickness,thestresscomponents
z,
zxand
zy
maybeconsideredaszerowithinthewholeplate:
z=0
zx=0
zy=0
zx
z=+t/2=0
zy
z=+t/2=0
z
z=+t/2=0Notingtheabsenceofsurfaceforcesonthefacesoftheplate,wehave:Theremainingstresscomponents
x,
yand
xyforthesamereason,maybeconsideredtobefunctionsofx
and
y
only:
x=f1(x,y)
y=f2(x,y)
xy=f3(x,y)Suchaproblemiscalledaplanestressproblemandthethinplateistobeinaplanestresscondition.yzxa.Theshapeofthebody::Verylongcylindricalorprismaticalbodyb.Mannerofexternalforces:(1)beparalleltoacrosssection(2)don’tveryalongtheaxialdirection2)Pl合an蜻e购st碍ra抬in(2)Du最e追to袜s仪ym信me安tr缴y呢(e阅ve效ry泊c北ro烂ss肌s捉ec机ti恢on和b暑ei杜ng周a却p舱la滑ne交o灰f听sy圈mm塑et厨ry丹),缠t浙he适s剑he受ar封in辟g即st漂re刚ss动es
zxan倚d
zymu晌st世b钳e柔ze晴ro达:
zx=0
zy=0(1半)校If胳t稼he暂b蝇od区y众we昆re弃o废f愿in速fi消ni骗te右l况en鹊gt锻h,泄a骨ll恒c牲om民po染ne蚕nt贷s兰of猫s乌tr所es盏s,伞s助tr巧ai团n资an粥d虚di乓sp渴la映ce萝me彩nt俗w偶ou摧ldno冤t糊va顽ry宣i魄n续th绿e姓ax驼ia凤l它di做re苗ct缝io租n.Wi疏th禾a馋ny津c欺ro斥ss秆s疑ec修ti井on棒o偷f批th锄e群bo课dy草a成sxypl冬an红e,裁t凭he坏c愧om施po坟ne旋nt央s粗wi侨ll佛b拦e枕fu倍nc觉ti必on腰s美ofxan攀dyon医ly可.Fo乓r卫ex岸am堵pl逝e:
x=f1(x竖,y府)
y=f2(x呈,y捎)
xy=f3(x捎,y鸡)(3筹)Th扫er碎e老wi鸡ll告b拾e渔no润rm她al认s阀tr驳es受s
zon船a用c挑ro压ss闯s蛙ec州ti刻on抱d怀ue终t单o释th劫e爸mu蚂tu摩al来c萄on剪st护ra币in毫t故be见tw纱ee病n跃th缩慧e春ma登te技ri萌al傲s晚on盒t页wo限s像id像es额o畅f司th项e萝cr某os颗s泊se絮ct从io废n.(4疤)Al坛so枝d旧ue应t篇o唯sy识mm良et骗ry竖,雨an祥y羽po问in枕t争of呀t誓he饭b即od境y脾wi首ll晴n跳ot叔m侍ov咽e恢in议t窝he郊a满xi顺al料d吵ir片ec绑ti认on什,揭an旗d流we叉h夏av增ew=0.Su度ch餐p王ro撒bl搭em想s顺ho迎ul递d迫ha剃ve定b洋ee捐n晴ca体ll幼ed柔apl幕an赵e困di兆sp鼠la荐ce触me迹nt踪蝶p闯ro姿bl领em.制Bu剧t绢tr练ad她it啄io替na墙ll蒜y,怎i渠t袄ha往s丘be镇en价c踢al绕le赢d朴apl置an把e师st既ra洲in微p牙ro钞bl亿em.xyOIsolateanelementintheformofarectangular.
t=12.践2DI织FF含ER钩EN惜TI商AL耳E移QU眨AT婚IO宗NS怀O崇F扁EQ晒UI愁LI丑BR丛IU联M
y
yx
xy
xcXYCo雹ns谱id血er誓t录he酷e退qu属il汪ib喷ri观um猛.
y
yx
xy
xXYxyoXYc
MC=0(1)
xy=
yx
X=0(2)
Y=0(3)foraplaneproblemIn姜t贺he维se突t只wo盼e贼qu散at商io似ns屋,滩th那er涨e甜ar绢e蔬th已re谷e袜un糟kn狸ow钻n淹fu谊nc樱ti料on想s.咱H盐en酸ce孟,鉴th卵e汁el趁as肝ti路ci座ty议p脆ro驴bl毅em鸟i烛sst签at子ic遗al搏ly姜i柔nd疏et统er底mi糕na帐te.矛To欧s芝ol伐ve去f睛or照t则he窄u槐nk浮no起wn养s杰tr摩es芒se民s,警w晌e例ha恶ve尸t糕o金co披ns赤id绘er螺t袜he鹅s蛾tr问ai敏ns痒a馆nd抛d肚is番pl汗ac鸟em丙en贵ts过.In闯a翼p亩la所ne纸s侵tr界ai丈n获pr连ob锁le扛m,振t页he夹n抓or淡ma则l蓝st滚re咏ss杆es
zar接e谨in狡e竞qu肆il之ib专ri钻um狮t概he胁ms细el治ve网s档an苍d观do释n建ot嫩a络ff惠ec稻t后th凶e完fo兆rm毛ul战at腹io丧n恨of评a酷bo饰veEq谋s..Wh六en烈t芒he完s转tr被es凝s糕co痰mp想on闻en仇ts
x,y,xyat谨a叮c尼er冤ta石in棉p哈oi裁nt容P纵a银re磨k求no呼wn舰,侵we乡丰c废an委c洽om竹pu丽te扣t我he叨s云tr竞es改s钥ac教ti烦ng欣o危n州an咐y川pl市an喉e枯pa郊ss犹in院g客th伏ro溜ug串h素th业is对p冷oi慰nt猎,剧pa妙ra涨ll谊el稻t携o桥th风e当z问ax汽is邀b宵ut枪i锅nc风li干ne借d卷to抓t察he融x身a晌nd菜y箱a虏xe溉s.2.浙3ST台RE丹SS晕A亲T房诚A姜PO得IN化T.甲P当RI番NC崖IP缓AL煌S缩慧TR欲ES隆SE特S
y
yx
xy
xXYxyo
xy
x
y
yxTa循ke碗a巷p悬la呢ne屋A成B冬pa痒ra党ll优el改t兽o丈th店e郑in骄cl荐in评ed票p循la振ne恨c奖on干si绞de虎re妖d薪an非d四at爷a伞s阔ma梦ll竹d陪is查ta村nc辰e纱fr粱om注P铜.花t羞=1PA乐=dxPB润=dyAB肾=ds
y
yx
xy
xxyoBAPsYNXN
N
NNTh晶e茅co叔si势ne犯s券of葬t棚he仁a暂ng马le灶s弊be谈tw折ee芳n秀th老e弯ou看tw衬ar恐d屿no瞎rm暗al夏t脉o开th闹e梨pl飘an适e较AB旦a公re:co邮s(N共,x瓶)=lco病s(N乒,y造)=mSo治:PA银=m·dsPB篮=l·dsCo位ns检id要er庸t欣he钟e柜qu饭il莫ib拒ri盗um近c恭on坦di士ti拣on鼻s:
X=财0
y
yx
xy
xxyoBAPsYNXN
N
NN-
x·l·ds·1益-xy·m·ds·1扫+XN·ds·1员=0Wh篇y坦th贷e贝bo冲dy适f芳or准ce衣c汗om伍po跪ne厅nt储s团ca酸n饭be脊n江eg激le巧ct凝ed染?XN=l
x+m
xy
Y=宪0YN=m
y+l
xy
y
yx
xy
xxyoBAPsYNXN
N
NNLe能t主th适e蚊no弓rm专al映s晨tr年es冬s寇an景d隔sh却ea距ri浇ng紧s勉tr明es情s碧on店A请B先be
Nan铸d
N,pr粱oj页ec蓬ti姐on脉o喝fXNan去dYNon牺a盏nd京p向er设pe堵nd西ic腿ul概ar阔t励o躺th校e愁no启rm帖alNwi恐ll桌g堤iv旬e
N=lXN+mYN
N=lYN-mXNTh年us胃,碰th幼e贝no惊rm萝al柏a节nd蚕s伞he樱ar领in精g姿st耗re洒ss仿o知n逐an剃y群pl痒an钻e枕de革fi桃ne瓶d蜂by似t犯he屈d蛮ir擦ec舍ti司on东c摸os纤in滤es罢c刮an品e肌as为il悦y劣be辈c营al圣cu时la选te首d,史p梅ro足vi断de佣d董th核at合t凶he挨s誉tr省es液s扬co剪mp勇on波en看ts
x,yan栏d暴
xyat变t但he犬p雅oi蹈nt靠P渠a驾re军k育no锦wn叙.Us袭in堤g略th怀e祥ex潜pr性es龙si嚷on咬s声of障XNan没d番YN,we屿o芒bt搅ai掩n
N=l2
x+m2
y+2lmxy
N=lm(y-x)+(l2-m2)xyxyoBAPYNXN
NXN=l
x+mxy=l
YN=m
y+lxy=m
Ifth岭e辞sh彻ea恼ri遍ng届s酬tr振es仪son屿a原c算er摩ta步in幕p普la义ne炼t后hr容ou维gh微Pva皱ni僚sh跑es,筛th衣at简p宽la躬ne斜i松s策ca搜ll令eda仪pr改in辣ci退pa持l遭pl屈an授eat见P张a众nd递t匹he择n糠or其ma得l怠st遮re塘ss鲁o测n承th郑e希pl枝an色e君is铺c打al即le曲da笔pr逢in薪ci寸pa扎l伤st哈re奋ssat榆P巷.Su鬼pp琴os只e躁AB淋i旋s算a串pr渐in郊ci摄pa伙l州pl庆an航e.
2-(x+y)+(x
y-xy2)=0Si脂nc嚷e析th寺e方ex两pr冰es佣si底on感w廉it旋hi难n慈ea昌ch桶s悼ig真n妇of茄s行qu网ar精e宁ro钓ot旁i启s镇al珍wa精ys稼p缝os当it效iv毛e,
1an嗓d崇
2mu皂st今b仆e蛙re毫al代q供ua胶nt益it别ie路s.呈Th猫is窗s捏ho具ws芳t顾ha破t球tw类o粪pr所in社ci旧pa吹l正st晶re毛ss化es敲e朽xi隙st跪a盲t爷th方e停po导in糟t顺PAn叔d:
1+2=米
x+yLe和t
1be妨t裤he寄a渴ng拖le痛b功et系we罗en
1an正dx.碌T棍he拔n,Si河mi妹la教rl害y,月w宜e逝ca激n迎ob集ta递ini.e.Be惨ca砖us刷e
1+2=士
x+y
2=x+y-
1ThenWecanobtainTh针at量e配xp恨re慢ss剩io删n踪蝶sh蜓ow爱s撑th脑at缩慧t傍he津t廊wo允p泉ri焦nc察ip态al趋s组tr派es溉se挖s灵ar贺e盟pe平rp信en惩di周cu尽la帜r雕to标e标ac鬼h热ot尤he感r.xyO
y
yx
xy
x
xy
x
y
xy
1
1
2
2Wh鉴en找t宋he棚p迅ri罩nc验ip挤al伴s定tr杯es消se削s拢at饮a熄p腹oi攀nt娃a郊re瘦k闻no仅wn锣,肃it逃i莲s缩慧ve罪ry肆e翻as衣y哨to梢d诊et嗓er装mi艺ne驱t目he资m常ax嫩im谈um静a爷nd斯m络in丧im砖um蒙s广tr独es蔽se隔s肯at伞t泡he钓p据oi忌nt减.Fo孝r而si够mp界li请ci甘ty籍o涨f皱an总al杜ys壮is摘,避th袄exan肃dyax饥es貌a担re撤p往la浪ce久d浅in油t坝he闹d初ir抓ec野ti革on县s替of
1an炼d
2,re俩sp稿ec蓝ti躲ve皂ly园.xyxy
1
1
2
2Th鹿us虫,良we国h尖av孝e
1=x
2=y
xy=0Th乘e胆no遣rm绑al扯s粮tr悔es乳s按on骆a遍ny蚂i色nc不li昼ne数d割pl呀an际e猛wi赚ll经b享e:xy
1
1
2
2
N=l2
x+m2
y+2lm
xy=l2
1+m2
2Th撕e纳re芳la极ti理on搬o宋flan透dmisl2+m2=1So滤,搏we孔c膏an壶o商bt林ai倘n
N=l2
1+(芳1-l2)2=l2(霉
1-筑
2)+桨
2Wh扯enl2=1,
Nis糟m肾ax脚im竞um菠.Wh盟enl2=0,
Nis弱m爽in率im炕um奸.Whenl2=1,
N=max=1Whenl2=0,
N=min=2Themaximumandminimumvaluesof
Nare
1and
2,respectively.Theshearingstressontheinclinedplaneis
N=lm(y-x)+(l2-m2)xy=
lm(2-1)=
NWhen,theshearingstress
Nreachesitsextremevalues.Th秃is握s拒ho体ws应t屈he探e滔xt芝re鸟me趁s肚he档ar皆in稳g平st贸re锯ss炉es垃a巩re劲a简ct广in息g户on柄p规la篮ne峡s迎in误cl瓶in草ed厅a吩t寄450wi握th炊t报hexzan永dyzpl嚼an片es洒.xy
1
1
2
2
N
N2.悟4GE领OM弄ET灶RI恳CA按L粱EQ散UA钟TI湿ON刮S.凉R括IG雪ID关-B兰OD竖Y禁DI码SP连LA掩CE犹ME炕NT探SA’B’P’Taketwolineelements:PABxyoPA垃=dxPB扮=dyuv
x:n届or缠ma丹l兔st舞ra俘in落o卸f奸PA
y:n灭or丛ma春l熔st帅ra云in堪o速f骆PB
xyor
yx:th筋e芝ch瓦an崖ge机o乖f吴th锋e妥ri粱gh为t票an呜gl踪蝶e英be显tw选ee便n贡PA杠a新nd池P虽B.姻T溜he轧c显ha还ng握e余is+泳.A’B’P’uvPABxyo
Th井en匀,丝式we农h婚av堆e贴th秤e窜fo鞭ll唤ow酸in膏g森th布re痰e贯ge旁om岩et姥ri耕ca似l师eq历ua深ti痰on呀s:Fo齿r岂a内gi尘ve万n裕se壳t晨of泡d峡is晕pl摆ac岩em调en峡t云co槐mp密on控en狂tsuan盼dv,挎as兵d粮ef恢in并it岭e楚fu午nc漂ti免on滨s倍ofxan改dy,蓬th违e蔬st绑ra细in斥c拆om央po职ne魄nt恐s象ar皂e搁de巾fi凝ne德d笨by撒t统he岔se是r页el喜at颈io劫ns袜.wh乓ic芽h蒜ex仓pr荒es嗽s银th客e魔re芦la续ti吸on织s间be箩tw玩ee丧n坛th烈e劫st副ra远in件c平om惯po笨ne索nt销s撕an闭d蛙th死e迷di财sp棚la蚀ce帅me氧nt校c刘om宫po沸ne滤nt仰s武in日a知p蜻la俭ne颈p悼ro李bl竞em睡.Fo治r朋a墓gi百ve颜n忍se悦t摩of嫩s怪tr揉ai都n风co趴mp报on佛en石ts劲,伪as否d把ef宵in泥it镇e谷fu耍nc棋ti阵on爬s妥ofxan鱼dy,葬ho捐we现ve削r,步t痒he宽d烟is偷pl餐ac黄em蜜en易t附co烂mp塞on敲en亲tsar孤e销no宜t猴wh遭ol锹ly舟d热et色er碌mi钱na知te.Le修t
x=0伟,锹
y=0,xy=0u=f1(y)
v=f2(x)
u=f1(y)=u0-
y
v=f2(x)=v0+
x
Th贡es捕e重ar辣e坐th德e无di问sp渣la版ce浸me领nt油c捏om姻po钩ne对nt透s愤co燃rr例es分po旱nd农in移g扑to笑z轧er不o柏st苍ra仓in然s醉an谣d刃be刘t望he客r野ig蛙id唤-b周od稿y贯di迷sp报la补ce乘me疤nt燃s.u0an其dv0ar跟e举th赛e疮ri严gi裁d-深bo师dy界t律ra旱ns赖la速ti秤on背s怀in系t册hexan勾dydi饿re乡丰ct袍io爬ns夜.
is非t勾he饰r接ig目id栋-b壤od崖y逗ro醉ta晌ti尖on塘.渐Th嗓is滩c瘦an礼b懂e劈燕il价lu壶st济ra之te杰d笋as旨f枣ol淹lo剩ws极.TheresultantdisplacementisWhenonly
isdifferentfromzero,thedisplacementcomponentsofanypointofthebodyare:u=-yv=x
Pxxyyr
roLe帅t点th句e签an现gl停e苦be都tw客ee畜n久th乌e欠di老re府ct驼io难n欠of体t馋hi你s胖re冰su辱lt艰an蛾t目di贱sp舅la弯ce饱me挥nt窑a驶nd积t秆he亲y迟a摘xi乌s兴be,从t姓he介n占we弓h遍av蚂eTh碑is辣s许ho饿ws星t坊ha掏t遭th纠e狐re捐su帐lt许an耍t横di煎sp活la浸ce跑me词nt或o蝇f雄P恩is将p家er系pe幕nd绕ic掏ul殃ar旋t赠o罪th庆e贞ra腊di坛al与l兴in悟e妇OP宾a庄nd毕c引on民se掉qu斩en纸tl萍y率al茶on煮g违th谱e虫ta速ng动en等ti骡al腔d鸣ir细ec蝇ti温on状a纵t沉P.Th峰e院co轻ns皮ta织nt
停is郑t夹he途r皆ig择id越-b烧od会y僻ro澡ta急ti乓on板o鸟f答th结e冬bo丈dy瓦a侵bo非ut斜t蛛he派z哈a沙xi捎s.Pxxyyr
ru=
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