材料力學(xué)第二章

第二章拉伸、壓縮與剪切第二章答案求圖示各桿指定截面的軸力，并作軸力爭(zhēng)。a）440kN350kN225kN1解：20kNFR4321FR=5kNFR4FN4440kN3FRFN3FN4=FR=5kNFN3=FR+40=45kN225kN20kN120kNFN22FN11FN2=-25+20=-5kNFN1=20kN45kN5kN

20kN5kN10kN110kN26kN36kN（b）1—1截面：123FN1=10kN10kN12—2截面：FN1110kNFN2=10-10=010kN2FN23—3截面：2FN3=6kNFN336kN310kN6kN圖示一面積為100mm200mm的矩形截面桿，受拉力F=20kN的作用，試求：（1）6的斜截面m-m上的應(yīng)力；（2）最大正應(yīng)力max和最大剪應(yīng)力max的大小及其作用面的方向角。解：F20103mP1MPaFFA0.10.2mcos213o0.75MPa30430o2sin60o130.433MPa22max1MPamax

2

0.5MPa圖示一正方形截面的階梯形混凝土柱。設(shè)重力加快度

g=

9.8m/s2,

混凝土的密度為2.04103kg/m3，F(xiàn)=100kN，許用應(yīng)力

2MPa

。試依據(jù)強(qiáng)度條件選擇截面寬度

a和b。F解：2.041039.82104N/m2mFNN1P4a2F24NP4a,1F1F[]aN1A1a2mPF1001034ba0.228m[]4210642104N23PF4a24b2FN23100103210440.228221044b2N2[],b304.161030.398m398mm2FN2106104b2242在圖示桿系中，AC和BC兩桿的資料同樣，且抗拉和抗壓許用應(yīng)力相等，同為。BC桿保持水平，長(zhǎng)度為l，AC桿的長(zhǎng)度可隨角的大小而變。為使桿系使用的資料最省，試求夾角的值。解：FN2FN1FYFXA21VV1V2

CFF0,FN2sinF0,FN2Fsin0,FN2cosFN10,FN1cosFsinFN2N2cos1FN1FN1PF,A12PsinAsinA2lFl(1cos)A1lcossincossinFl2cot)(tan(1sin2cos2tancot)cossincossin由dV0,(tan)1,(ctg)1dcos2sin2d(tan2ctg)120dcos2sin2sin22cos20,sin22cos20sin2cos2tan22,tan2,54.44圖示桁架ABC，在節(jié)點(diǎn)C承受集中載荷F作用。桿1與桿2的彈性模量均為E，橫截面面積分別為A1=2580mm2,A2=320mm2。試問(wèn)在節(jié)點(diǎn)B與C的地點(diǎn)保持不變的條件下，為使節(jié)點(diǎn)C的鉛垂位移最小，應(yīng)取何值(即確立節(jié)點(diǎn)A的最正確地點(diǎn))。解：FN1Fcot,FN2F/sinl1FN1l1FcotlEA1EA1l2FN2l2FlFEA2EA2sincosFN2l1l2l1Fl1cotl2CCVtanEsin2cosA2A1tanFN1sindCV0d8cos30Fddsin2cos55.7o圖示桿的橫截面面積為A，彈性模量為E。求桿的最大正應(yīng)力及伸長(zhǎng)。FAF

l

l

Fxl2EA

0

dxEAFlFlFl2EA2EAEA圖示硬鋁試樣，厚度2mm，試驗(yàn)段板寬b=20mm，標(biāo)距l(xiāng)=70mm，在軸向拉力F=6kN的作用下，測(cè)得試驗(yàn)段伸長(zhǎng)的彈性模量E與泊松比。

l0.15mm，板寬縮短b0.014mm，試計(jì)算硬鋁解：FNl600070E70000MPal200.15EA2Eb/l0.014/0.150.327bl2070圖示一階梯形截面桿，其彈性模量E=200GPa，截面面積AI=300mm2，AII=250mm2,AIII=200mm2。試求每段桿的內(nèi)力、應(yīng)力、應(yīng)變、伸長(zhǎng)及全桿的總伸長(zhǎng)。解：15kN10kN25kN30kNFN130kN,ⅠⅡⅢ11.5m2mF11m301031NN1100MPaA1300106111001060.51030.05%E200109ll0.510310.05mm111NN215kN,FN2151032260MPaF2A225010622601060.31030.03%E200109l22l20.31031.50.45mmFN325103FNN325kN,3N3125MPa3A3200106331251060.6251030.0625%E200109l33l30.62510321.25mml3l1l2l30.50.451.252.2mm圖示一三角架，在結(jié)點(diǎn)A受鉛垂力F=20kN的作用。設(shè)桿AB為圓截面鋼桿，直徑d=8mm，桿AC為空心圓管，橫截面面積為40106m2,二桿的E=200GPa。試求：結(jié)點(diǎn)A的位移值及其方向。解：FN5F=25kN,FN3F=PNP15kNAB44FNABl251032.5lABABAB826.22mmEA200109106AB4F1531.5lACNl102.81mmEAAC20010940106xAlAC2.81mmyAlAB5lAC39.88mm44AA'10.3mm

B2.5mAC1.5mFFNABFNACAFlACAlABA′圖示一剛性桿AB,由兩根彈性桿AC和BD懸吊。已知：F,l,a,E1A1和E2A2，求：當(dāng)橫桿AB保持水平常x等于多少解：FN1lFx0FN1xP1lCE1A1FN2lF(lx)0F(lx)aN2P①N2lAxl1l2,lFN11l1FNl2FN1N22,E1A1E2A2AxlxPxPE2A2lll,xE1A1E2A2E1A1E2A2一剛性桿AB,由三根長(zhǎng)度相等的彈性桿懸吊?！?、○2、○3桿的拉壓剛度分別為和E3A3,構(gòu)造受力如下圖。已知F、a、l，試求三桿內(nèi)力。

DE2A2②BFFN2BFE1A1、E2A2A11A2A3③lE①E2②E3解：AaaBFY0FN1FN2FN3F0FMB0FN22FN10FN1FN2FN3l1l32l2ABFN1lFN3l2FN2lFE1A1E3A3E2A2l1l2l3FN1FFN1FN1N1PN12N1E1A1E3A32E2A2F2FPE3A32PFNN1=,NE3A314FN2=1411E1A1E2A2E3A3E1A1E2A2E3A314FN3=3E1A1E2A2F141E1A1E2A2E3A3橫截面面積為A=1000mm2的鋼桿，其兩頭固定，荷載如下圖。試求鋼桿各段內(nèi)的應(yīng)力。解：FRA+FRB–100–150=0幾:l1l2l3FN1=FRA，F(xiàn)N2=FRA-100，F(xiàn)N3=FRBFRA0.5(FRA100)0.3EAEA

FRAAFRAFRA500300100kN100kNFN3FN1400150kNFN2BFRBFRBFRB0.40EAFRAA0.5(RFRAA100)0.3FRRB0.40,FRB=2FRA-75FRA+FRB=250，F(xiàn)RA=108.3kN，F(xiàn)RB=250-FRA=141.7kNFN1=FRA=108.3kN（拉力）FN2=FRA-100=108.3-100=8.3kN（拉力）FN3=FRB=141.7kN（壓力）FNN1FRRA108.310311A4108.3MPaAA1010FN2FRA1008.3103N2RA8.3MPa2AA10104N3RB141.71033FN3FRB141.7MPaAA10104木制短柱的四角用四個(gè)40404的等邊角鋼加固。已知角鋼的許用應(yīng)力鋼＝160MPa,E鋼＝200GPa；木材的許用應(yīng)力木＝12MPa,E木＝12MPa。試求允許荷載F。FF解：FNm+FN=F△Lm=△LGGNmFFNmmFNmNNgg,EmAmFNgNNmN,1EmAmEgAgFmEgAgFNgNNgFP[]gAg0FgEmAm15EgAg2250[P]1[]gAg(1EmAm)FEgAg643.086104(1121090.250.254)798kN16010943.0861020010NNgEgAgN,FgFNmEmAmFPNFNmEgA1EmA

[]mAmgm[PF]2[]mAm(1EgAg)EmAm1210640.250.25(120010943.086104)997kN12109[FP][PF]1798kN2-14

在圖示構(gòu)造中，

1、2兩桿的抗拉剛度同為

E1A1,3桿的抗拉剛度為

E3A3,長(zhǎng)為

l。在節(jié)點(diǎn)處受集中力F。試求將桿1、2和3的內(nèi)力。FBCD解：FX0,FN2sinFN1sin0132FN3Fy0,FN2cosFN1cosFN3F0FN1FN2l2FN1cosFFN3Al1l2l3cosFFFN1l1FN3l1cosFN1E1A1cos2FN3l1,E1A1l3E3A3E3A3BCD132FN1FN2cos2FE3A32cos3⊿l1E1A1⊿l3FFN32E1A1cos31E3A3求圖示聯(lián)接螺栓所需的直徑

d。已知

P=200kN，

t=20mm

。

螺栓資料的[τ]=80Mpa,[bs]=200MPaσ。解：FF/2t/2FSFP[],FV22PtAd2d2F/2t/24d12PF220010340mm[]80106FbsF3FP20010PbsPbs],d2650mmAbs[t[bs]td0.0220010dd250mm圖示元截面拉桿，承受軸向拉力F,已知0.6，試求拉桿直徑d與端頭高度h之間的合理比值。DSFh[]0.6[]解：dAVdhFPFFF4PP[]ALd2d24FPd0.6[]ddh4FP4h[]2.4h