材料力學(xué)(柴國(guó)鐘、梁利華)第6章

材料力學(xué)(柴國(guó)鐘、梁利華)第6章6.1 用積分法求圖示各梁的撓曲線(xiàn)方程及自由端的撓度和轉(zhuǎn)角，設(shè)梁的剛度EI為常數(shù)。解：（a）Mx1Fx1,x10,ax1x22FABMx2F2ax2,x2a,2aaCaF由EIwxMx，可得當(dāng)x10,a時(shí)，EIwx1Fx1，EIx1Fx12Fx13D1C1，EIwx1C1x126當(dāng)x2a,2a時(shí)，EIwx2F2ax2，F(xiàn)4ax2x22C2，EIx2F6ax22x232C2x2D2EIwx26邊界條件：當(dāng)x10時(shí)，0w00，代入，即得C1D10連續(xù)性條件：當(dāng)x1x2a時(shí)，1a2a，w1aw2a，代入Fa23Fa2C2C2Fa2；Fa35Fa3C2aD2D2Fa322663因此，梁的撓曲線(xiàn)方程為：wx1Fx13,x10,a6EIx236ax226a2x22a3wx2Fa,2a6EI,x2梁的轉(zhuǎn)角為：x1Fx12,x10,a2EIx2Fx224ax22a2,x2a,2a2EI自由端的撓度和轉(zhuǎn)角為：wBFa3Fa2EI，BEIMx1122ax1,x10,aqq3aAB（b）2Mx21q2ax22,x2a,2ax1x22aa由EIwxMx，可得當(dāng)x10,a時(shí)，2EIwx11qa3a2x1，EIx11qax13ax1C1，22EIwx1129a2x1C1x1D112qax1當(dāng)x2a,2a時(shí)，EIwx21q2ax22，EIx2qx212a26ax2x22C2，26EIwx2qx2224a28ax2x22C2x2D224邊界條件：當(dāng)x10時(shí)，0w00，代入，即得C1D10連續(xù)性條件：當(dāng)x1x2a時(shí)，1a2a，w1aw2a，代入3317qa44qa37qaC2C2qa；7qa4C2x2D2D2qa24661224因此，梁的撓曲線(xiàn)方程為：wx1qax129a2x1,x10,a12EIwx2qx248ax2324a2x224a3x2a4a,2a24EI,x2梁的轉(zhuǎn)角為：x1qax13ax1,x10,a2EIx2q12a2x26ax22x23a3,x2a,2a6EI自由端的撓度和轉(zhuǎn)角為：wB41qa4，B7qa324EI6EI（c）Mx3q01q0lx6l由EIwxMx，可得AxlB1lx3，EIwxq0l61l4xC，EIxq024l5EIwx1q0llxCxD120邊界條件：當(dāng)x0時(shí)，0w00，代入，即得Cq0l3，Dq0l424120因此，梁的撓曲線(xiàn)方程為：3q0lx54xl55lEIwx120lEI梁的轉(zhuǎn)角為：q0lx44lEIx24lEI自由端的撓度和轉(zhuǎn)角為：q0l4， q0l3wB B30EI 24EId）Mx11qx12,x10,a2Mx2qax23qa2,x2a,2a2由EIwxMx，可得當(dāng)x10,a時(shí)，

x2q qa2A x1 Ba aEIwx11qx12，EIx11qx13C1，EIwx11qx14C1x1D12624當(dāng)x2a,2a時(shí)，EIwx2qax23qa2，EIx21qax223qa2x2C2222EIwx21qax233qa2x22C2x2D264邊界條件：當(dāng)x22a時(shí)，2aw2a0，代入，即得C2qa3，D21qa43連續(xù)性條件：當(dāng)x1x2a時(shí)，1a2a，w1aw2a，代入，即得C11qa3，D15qa4624因此，梁的撓曲線(xiàn)方程為：qx144a3x15a4,x10,awx124EIqa2x239ax2212a2x24a3a,2awx212EI,x2梁的轉(zhuǎn)角為：4qa3x13,x10,ax16EIqax22ax2a,x2a,2ax22EI自由端的撓度和轉(zhuǎn)角為：5qa4，qa3wBB24EI6EI6.2 用積分法求圖示各梁的撓曲線(xiàn)方程、截面A和截面B的轉(zhuǎn)角、跨度中點(diǎn)C的撓度以及外伸端的撓度與轉(zhuǎn)角，設(shè)梁的剛度EI為常數(shù)。

x2x1ABqa/4C3qa/4aa(a)解：（a）Mx1,x10,aqax14q2x225ax22a2a,2aMx24,x2由EIwxMx，可得當(dāng)x10,a時(shí)，EIwx1qax1，EIx1qax12C1，EIwx1qax13C1x1D14824當(dāng)x2a,2a時(shí)，22，EI322，EIwx2q2x25ax22ax2q4x215ax212ax2C2424EIwx2qx245ax236a2x22C2x2D224邊界條件：當(dāng)x10時(shí)，w00；當(dāng)x22a時(shí)，w2a0，代入，即得D10，2aC2D20連續(xù)性條件：當(dāng)x1x2a時(shí)，1a2a，w1aw2a，代入qa3C1C2，qa4aC1aC2D268334聯(lián)立求解，可得：C17qa48，C2qa48，D2qa24。因此，梁的撓曲線(xiàn)方程為：5wx1qa2x137a2x1，wx2q2x2410ax2312a2x22a3x22a448EI48EI梁的轉(zhuǎn)角為：qa6x127a2x148EI

q8x2330ax2224a2x2a3，x248EI截面A和截面B的轉(zhuǎn)角為：A7qa3，B3qa3，Cqa348EI16EI48EI跨度中點(diǎn)C的撓度為：wC5qa448EIq02l2x3lx2x3x（b）Mx6lq0BA由EIwxMx，可得Cq0l/3lq0l/6EIwxq02l2x3lx2x3，(b)6lEIxq04l2x24lx3x4C，24lEIwxq020l2x315lx43x5D360lCx邊界條件：當(dāng)x10時(shí)，w00；當(dāng)x2l時(shí)，wl0，代入，即得Cq0l3，D045因此，梁的撓曲線(xiàn)方程為：wxq020l2x315lx43x58l4x360EIl梁的轉(zhuǎn)角為：xq04l2x24lx3x4q0l324EIl45EI截面A和截面B的轉(zhuǎn)角為：Aq0l3，B7q0l345EI360EI跨度中點(diǎn)C的撓度為：5q0l4wC（c）2x2Mx1qax1x1,x10,2ax1qa2qq3a2ABMx2ax2,x22a,3aCDqa/22a5qa/2a6(c)由EIwxMx，可得當(dāng)x10,2a時(shí)，EIwx1qax1x12，EIx1q3ax122x13C1，21234EIwx1q2ax1x1C1x1D124當(dāng)x22a,3a時(shí)，EIwx2q3a2ax2，EIx2q6a2x2ax22C2，q9a2x22ax232EIwx2D26C2x2邊界條件：當(dāng)x10時(shí)，w00；當(dāng)x22a時(shí)，w2a0，代入，即得D10，C10，14qa42aC2D203連續(xù)性條件：當(dāng)x1x22a時(shí)，12a22a，代入聯(lián)立求解，可得：C10，D10，C211qa3，D28qa4。33因此，梁的撓曲線(xiàn)方程為：wx1q2ax13x14，wx2q9a2x22ax2311qa3x28qa424EI6EI3EI3EI梁的轉(zhuǎn)角為：x1q3ax122x13，x2q6a2x2ax2211qa312EI2EI3EI截面A和截面B的轉(zhuǎn)角為：A0，B5qa36EI跨度中點(diǎn)qa4C的撓度為：wC24EI為：2qa4wB

；外伸端B的撓度（d）Mqax14a2,x10,2ax2qx1qa2x142ABq3ax2M,x22a,3aCx22qa/43qa/42aa(d)7由EIwxMx，可得當(dāng)x10,2a時(shí)，EIwqax14a2，x14322qax112ax1C1x1D1EIwx124當(dāng)x22a,3a時(shí)，q3ax22EIwx2，2q3a4x2C2x2D2EIwx224邊界條件：當(dāng)x10時(shí)，w0代入，即得

qax128a2x1C1，EIx18q3ax23，C2EIx260；當(dāng) x22a時(shí)，w2a0，D10，C15qa3，2aC2D2qa4624連續(xù)性條件：當(dāng)x1x22a時(shí)，12a22a，代入C25qa36聯(lián)立求解，可得：C15qa3，D10，C25qa3，D241qa4。6624因此，梁的撓曲線(xiàn)方程為：qax1312ax1220a2x1，wx2q3ax245qa3x241qa4wx124EI24EI6EI24EI梁的轉(zhuǎn)角為：qax128a2x15qa3，x235qa3x1q3ax28EI6EI6EI6EI截面A和截面B的轉(zhuǎn)角為：5qa35qa3A6EI，B6EI跨度中點(diǎn)C的撓度為：43qa8EI為：wB19qa4。24EI

；外伸端B的撓度6.3用疊加法求圖示各梁截面A和截面B的撓度，以及截面C的轉(zhuǎn)角，設(shè)梁的剛度EI為常數(shù)。82FFaqqa2BAAaaCaBaC(a)(b)qqa2qaqBCABACaaaaaa(c)(d)解：（a）梁截面A的撓度為2Fa32232FaFa2awAFBMAFBFBMAFawAwAwBBawA3EI2EIa2EI3EI梁截面B的撓度為wBwBFBwBMA2Fa3Faa2Fa33EI2EI6EI梁截面C的轉(zhuǎn)角為2Fa2Fa22Fa2Fa2FBMAFBMAFa2aCCCBC2EIEIEIEIEI（b）梁截面A的撓度為42wAwAqwAMAq2aqa22a08EI2EI梁截面B的撓度為wBqMAqa2a28a224a2qa45qa4wBwB24EI2EI24EI梁截面C的轉(zhuǎn)角為qMA8qa3qa22a2qa3CCC6EIEI3EI（c）梁截面A的撓度為5q2a4122wA2qa2aqa4384EI16EI12EI梁截面B的撓度為q2a31qa22aqa4qa4wBa2a3EI8EI8EI24EI梁截面C的轉(zhuǎn)角為q2a31qa22aqa3qa3C224EI3EI6EI6EI9（d）梁截面A的撓度為2a4a2a231qa22a25qa4qaqa2a2wA12EIa48EI16EI24EI梁截面B的撓度為qa22aaqa2a21qa22aaqa4qa4qa413qa4wBa6EI16EI23EI8EI3EI8EI24EI梁截面C的轉(zhuǎn)角為qa22aqa2a2qa37qa3C16EI6EI12EI3EI6.4用疊加法求圖示梁截面A的撓度，設(shè)梁的剛度為常數(shù)。解：因?yàn)閣AwA(a)wA(b)wA(c)，又wAwA(c)，q故AEIwA1wA(b)因?yàn)锳2A(a)15q2a4(b)wAwA2384EI2

5q45qa42a2384EI48EI，故AA，又AA(b)(c)(c)5qa448EI

a a2EI EIa a

FA6.5用疊加法求圖示變截面梁自由端A的撓度和轉(zhuǎn)角。解：wAFa3Fa2Faa2FaaFa33Fa332EI22EIa2EIa2EI22EI3EIAFa2FaaFa25Fa222EI2EI2EI4EI6.6 圖示結(jié)構(gòu)中，拉桿AC的橫截面面積為A2500mm2，彈性模量E1100GPa。梁BC的橫截面為200mm 200mm的正方形，彈性模量E2200GPa。試用疊加法求梁的中間截面D的鉛垂位移。

q=40kN/mBD1m 1m

A2mC10解：wD5qa4qal384E2I4E1A125402000440200020003842001032004410010325000.31250.160.4725mmqa6.7用疊加法求圖示剛架截面A的水平位移和鉛垂位移。設(shè)各個(gè)梁段的剛度EI為常數(shù)，拉壓變形忽略不計(jì)。

EI aqaA解：qa3qa2aqaa33qa4uAEIa2EI6EI3EI4qavA8EIy

F6.8 滾輪沿簡(jiǎn)支梁移動(dòng)時(shí)，要求滾 A輪恰好走一水平路徑。試問(wèn)需將l梁的軸線(xiàn)預(yù)先應(yīng)彎成怎樣的曲線(xiàn)？設(shè)梁的剛度常數(shù)。解：wxFlxxl2x2lx2Flxx26EIl3EI6.9橋式起重機(jī)的最大載