第三章薄板理論

1、February 4, 20221February 4, 20222February 4, 20223February 4, 20224bSDSFebruary 4, 20225( )( )uvzz000),()(0yxwwzxzyzzz上述假定統(tǒng)稱為克?；舴颍ㄉ鲜黾俣ńy(tǒng)稱為克?；舴颍↘irchhoff）假定。）假定。彈性薄板小撓度理論的基本假設彈性薄板小撓度理論的基本假設February 4, 20226rzqqqxFebruary 4, 20227MrMQrFebruary 4, 20228Fx0Fy0Mx0Mz0Fz0Mx0February 4, 20229d()d( )rQrrq rr

2、d( .)d.r MrMQ rrr0qqqxFebruary 4, 202210Fz0QQrrrrQ rq r rrrrrdddddd( ) d d0Mx0MMrrrrM rMrQ rrq r rrrrrrrddddddd sin(d)d d( ) d dd2220qqxFebruary 4, 202211Fz0QQrrrrQ rq r rrrrrdddddd( ) d d0QrQrrq rrrdd( )d()d( )rQrrq rrMx0MMrrrrM rMrQ rrq r rrrrrrrddddddd sin(d)d d( ) d dd2220MrMrMQ rrrrdd0d( .)d.r

3、MrMQ rrr0sind2d2February 4, 202212（a）圓板中面的變形 （b） （a）中部放大圖圖3-3 受軸對稱載荷圓板的幾何變形February 4, 202213r 0rwddFebruary 4, 202214 dzzdr)d(rrzzrrzrd(d)ddddFebruary 4, 202215r()rz2 ()rzrrzr22rwrzrwzrdddd22zrwddFebruary 4, 202216幾何方程：幾何方程：drdwrzdrwdzr22February 4, 202217z 0rrrEE11rrrEE1122幾何方程：幾何方程：drdwrzdrwdzr22

4、February 4, 202218rrrz2S222max222maxdddd1)1 (2)(dddd)2(1)(rwrwrESrwrrwESrFebruary 4, 202219 February 4, 202220rzzrzdddMz zrr2222ddSSrSSrrzzMM February 4, 202221 zddMz z2222ddSSSSzzMMdzFebruary 4, 2022222222dddd1ddddrwrwrDMrwrrwDMr)1 (1223ESDzSMzSMrr331212February 4, 202223圖3-5應力分布圖2Sz2max2max6)(6)(h

5、MhMrrDQrwrrwrrwrdd1dd1dd22233February 4, 202224DQrwrrrrrdddd1dd11rrrr rrrq rDdddddd( ) Drqrwrrrrrrr)(dddd1dddd1rwddFebruary 4, 202225February 4, 202226圓板軸對稱彎曲的小撓度微分方程圓板軸對稱彎曲的小撓度微分方程DQdrdwrdrdrdrdr1February 4, 202227求解圓平板彎曲應力的基本步驟求解圓平板彎曲應力的基本步驟外載荷外載荷P剪切內(nèi)力剪切內(nèi)力Qr圓平板軸對稱彎曲微分方程圓平板軸對稱彎曲微分方程彎曲內(nèi)力彎曲內(nèi)力Mr、 M彎曲應

6、力彎曲應力r、 February 4, 202228 qrq)(Qr22r Qr qrQqrr2Dqrrwrrrr2dddd1ddFebruary 4, 202229)216(dd213rCrCDqrrw32214ln64CrCrCDqrwC CC123,February 4, 20223002C)216(13rCDqrdrdw321464CrCDqrwDqrCDMDqrCDMr163112163122121r2222dddd1ddddrwrwrDMrwrrwDMr(3-13)(3-14)February 4, 202231 (a) -3/4（qR2/S2） (b) 圖 3-7 周邊固支實心圓

7、平板 R r q 0 0 rrRrRqRDC rCqRDC R4123316401620CqRDCqRD12343264 ,February 4, 202232CC13, ddrqrDRr162222264rRDqw DqRwwr6440maxrFebruary 4, 202233MqRrMqRrr1613161132222()()223223)31 (143)3(143rRSqzrRSqzrFebruary 4, 202234()()MMqRrrr0021612200143)()(RSqzrrrrR(),()MqRMqRrr Rr R 2288MMqRrr Rmax() 28222max43

8、)(SqRSzRrr323223)(,23)(SqzRSqzRRrRrrFebruary 4, 202235February 4, 202236 r R (a) (b) 圖3-8 周邊簡支實心圓平板 q 0 0 r228)3 ( 3SpRrR0wrRMr 0qRDC RCqRDC412321640163210CqRDCqRD123431325164 ,February 4, 202237221316rRDqrdrdwDqRrDqRqrw64153213644224r 0DqRwwr6415)(40maxFebruary 4, 202238MqRrMqRrr163163132222()()()2

9、23223)31 (343)(3(43rRSqzrRSqzrFebruary 4, 202239r 0()()MMqRrrr0021632300343)()(RSqzrrrrR(),()()MMqRrr Rr R081232)1 (23)(,0)(SqzRRrRrrMMMqRrrrmax()()002163)3(83)()(222020maxSqRSzrSzrrFebruary 4, 202240February 4, 20224108.415maxmaxfsww65.123maxmaxfsFebruary 4, 202242 薄圓平板應力特點薄圓平板應力特點 板內(nèi)為二向應力板內(nèi)為二向應力 、

10、 ，平行于中面各層相互之間，平行于中面各層相互之間的正應力的正應力 及剪力及剪力 引起的剪應力引起的剪應力 均可予以忽略。均可予以忽略。正應力正應力 、 沿板厚度呈直線分布，在板的上下表沿板厚度呈直線分布，在板的上下表面有最大值，是純彎曲應力。面有最大值，是純彎曲應力。應力沿半徑的分布與周邊支承方式有關，工程實應力沿半徑的分布與周邊支承方式有關，工程實際中的圓板周邊支承是介于兩者之間的形式。際中的圓板周邊支承是介于兩者之間的形式。 薄板結構的最大彎曲應力薄板結構的最大彎曲應力 與與 成正比，而薄成正比，而薄殼的最大拉（壓）應力殼的最大拉（壓）應力 與與 成正比，故在相同成正比，故在相同 條件下

11、，薄板所需厚度比薄殼為大。條件下，薄板所需厚度比薄殼為大。 rzrQrmax2SRmaxSRSRFebruary 4, 202243 0dddd1ddrwrrrrrCrCrw212ddFebruary 4, 2022443221lnCrCrCw)1 (12)1 (12221221rCCDMrCCDMrC CC123,February 4, 202245rR1MMr1rR0wMMr2C RCRCCCRMDCCRMD122312121122202 112 11 ln()()CM RM RDRR11122221221()()CMMR RDRR2122122121()()()CRD RRM RM RM

12、MR322121122212122 11 ()()()lnR February 4, 202246rRRMMrRMRMRRDrw)()()(dd1112122121122212)()(ln)()(2221122212212121211rRRMRMRrRRMMRRDwMRRrM RM RrR RMMMRRrM RM RrR RMMr11212222112221212212222112221212()()()()()(12)()(1221212221122322122121222112232212MMRRrRMRMSrRRzMMRRrRMRMSrRRzrFebruary 4, 20224701M0

13、2MrRrRRDRMrw)1 (1)(dd2212211)1 (2Rrln1)(222212211rRRRRDRMwMM RRRRrMM RRRRrr 112212221122122211()()()()1 ()(12)1 ()(12223212211223212211rRSRRzRMrRSRRzRMrFebruary 4, 20224802M01MrRrRRDRMrw)1 (1)(dd2121222RrRrRRRDRMwln1)1 (2)(212221222MM RRRRrMM RRRRrr222121222221212211()()1 ()(12)1 ()(1222132122222132

14、1222rRSRRzRMrRSRRzRMrFebruary 4, 202249MM 201MR10)1 (ddDMrrw)1 (2)(22DrRMwMMMr312SMzrFebruary 4, 202250QPrQr221rQR QPrQPrr2rDPrwrrrr2dddd1ddFebruary 4, 202251 dd( ln)rPrDrC rCr82121232212ln) 1(ln8CrCrCrDPrwMDPDrCCrr 82 11211122()ln()()()MDPDrCCr 82 11211122()ln()()()2222dddd1ddddrwrwrDMrwrrwDMr(3-42

15、)February 4, 202252rR0wMr 0rR1Mr 0PRDRC RCRCPDRCRPDRCR212231221121281082 11211082 112110(ln)ln()ln()C ()()()ln()C ()()CPDRRRRRR12121212812 1 ()lnlnCPDR RRRRR12212212411 lnCPRDRRRRRRRRRRR3221212121221218112 112 11()lnR lnln()lnFebruary 4, 202253 PrDrRRRRRRrR RRRRR282111212 112121222122121ln(lnln)()()

16、ln)1 (21181ln1)1 (2ln)(8lnln)1 (211ln821212212211122DPRRrRRRRDRPRRRRRRRrDPrw22MPrRRRRRRrR RRRRRr ()lnlnlnln141212121222122121MPDrRRRRRRrR RRRRR1 ()lnlnlnln1411121212122212212February 4, 202254 P r R 圖 3-12 圓板中心受集中載荷 R1RRR121lnCPDR1812 1 ()lnC20)1 (211823DPRC)()1 (23ln8222rRRrrDPwFebruary 4, 202255February 4, 202256PQP