![材料力学 Ⅰ](https://wfqqreader-1252317822.image.myqcloud.com/cover/525/654525/b_654525.jpg)
习 题
基本题
2-1 阶梯杆如习题图2-1所示,AC段为圆形截面,其直径d = 16 mm,其上有一直径d =3 mm的小通孔,BC段为矩形截面,其截面尺寸b × h = 11×12 mm2,试作轴力图并计算杆横截面上的最大正应力。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0056_0004.jpg?sign=1734396067-hB8VOLQTitHQyIazF8cgYLPSfPJohvTA-0-ddca36cc210d93a37e7864ba49784410)
习题图2-1
2-2 如习题图2-2所示的直杆,横截面面积A =100 mm2,载荷F = 10 kN,试求α=−60°斜截面上的正应力和切应力。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0056_0005.jpg?sign=1734396067-GQ8j3gRudsZYsp0PWWwWFlVJO5fZPRqF-0-db5a237de4e5c3aa063c2790fcffa5b3)
习题图2-2
2-3 图示链条由形状相同的钢板铆成,t = 4.5 mm, H = 65 mm,h = 40 mm,d = 20 mm,材料的许用应力[σ] = 80 MPa,若F = 25 kN,试校核链条的抗拉强度。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0056_0006.jpg?sign=1734396067-85JR7dZkCSWduzVYYWMcNpu1JakVRv3B-0-5f9eadc9bc592567f4722ad2d2843c01)
习题图2-3
2-4 图示吊环,由斜杆 AB、AC 与横梁 BC 组成。已知α = 20°,F = 1200 kN,斜杆许用应力[σ] = 120 MPa。试确定斜杆的最小直径d。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0057_0001.jpg?sign=1734396067-DaUwCHtiBv0nyO4DSQdHvp4sAGNqY9Qh-0-6a81a08ed339a5b351b2417fffb7fbf1)
习题图2-4
2-5 图示结构杆AC、BC均为直径d = 20 mm的圆截面直杆,材料许用应力[σ] = 160 MPa,试求此结构的许用载荷。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0057_0002.jpg?sign=1734396067-alggFPlr6CVBQ52fNUBhm7oIbT1K2q41-0-f6200160e906fd5314d620c37d777829)
习题图2-5
2-6 已知一矩形截面拉杆在杆端承受拉力F=1100 kN。该拉杆橫截面的高度与宽度之比h/b = 1.4。材料的许用应力[σ]= 50 MPa,试确定截面高度h与宽度b。
2-7 悬臂吊车如图所示,杆CD由两根36 × 36 × 4等角钢组成,其许用应力为[σ]= 140 MPa,试校核杆CD的强度。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0057_0003.jpg?sign=1734396067-2vtJ0TwckhckfDB1xdCtqlG7stdk5okw-0-1f5c1fe9f99dfb4ea6c28598bd9ffbbc)
习题图2-7
2-8 两根直径不同的实心截面杆,在B处焊接在一起,弹性模量均为E = 200 GPa,受力和杆的尺寸等如图所示。试:(1) 画轴力图;(2) 求杆的轴向伸长量。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0057_0004.jpg?sign=1734396067-2uUtS81PORNOrnRUvVfwU0PI97zLbgsJ-0-263b03b0a2682ca8ec8094ab20133b12)
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0057_0005.jpg?sign=1734396067-IAm8lETHCuLXwTGMxsGpRgDNNjV4vDQK-0-45f62c0b2d6897a5de240c204b01022c)
习题图2-8
2-9 图示钢杆AB,已知F = 10 kN,L1 = L2 =400 mm,A1 = 2A2 = 100 mm2,E = 200 GPa。试求杆AB的轴向伸长量。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0057_0006.jpg?sign=1734396067-IpviMo7TSelSBCWqA4WFZpwGuCnF26HO-0-88c0b8c8407a50f0874ba710b940e05f)
习题图2-9
2-10 如图所示的结构,杆AB的重量及变形可忽略不计。杆1和杆2的弹性模量分别为E1 =200 GPa,E2 = 100 GPa,F = 60 kN,试求AB 杆保持水平时载荷 F 的位置及此时杆1、杆2横截面上的正应力。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0057_0007.jpg?sign=1734396067-Xg8MPfU2iX2irl76CXOafoUZXjz7nTgQ-0-1e9e911114ed56746cb41c154566be67)
习题图2-10
2-11 如图所示钢杆1、2的弹性模量均为 E =210 GPa,试求结点A铅垂方向的位移。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0057_0008.jpg?sign=1734396067-xUhrQxPQUjiy0Z8LLV6hAxjMMvob2dTz-0-d151fc53bdc1b8747b2fcb7e644c2bff)
习题图2-11
2-12 图示结构,AB为刚性杆,斜杆CD为直径d = 20 mm的圆杆,其材料许用应力[σ]=140 MPa,弹性模量E = 200 GPa,已知F = 15 kN,试用两种不同的方法求点B的铅垂位移ΔB。(提示:其中一种为能量法。)
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0058_0001.jpg?sign=1734396067-GzS4xrtjfDU0jOnd5vJs5w6ks2C7dhh6-0-ce18b1dbf6cf4429703be5481e9088ba)
习题图2-12
2-13 挂架由杆 AC 及杆 BC 组成,二杆的 EA相同,C处作用有载荷F。试求点C的水平及铅垂位移。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0058_0002.jpg?sign=1734396067-wYMCsbTezca6KRZFd03GTfAeSGuLQSiX-0-0ec17d8b5c4ddb9eb5eb6cde629ddc9c)
习题图2-13
2-14 图示结构,杆AB、杆AC和杆AD的横截面面积A、长度L、弹性模量E均相等,若外载为F,α= 45°,试求各杆的应力。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0058_0003.jpg?sign=1734396067-Lh0j2ttqd1MAwLf1tv64xztiplk8aWqo-0-d00b34aa0772d8ce1ea2a06c2b495255)
习题图2-14
2-15 试作图示杆AB的内力图。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0058_0004.jpg?sign=1734396067-YHxqdkyLlHdyagRsiQefezSloP6y5fU1-0-68ccee438a226708680b059c2ce64f3e)
习题图2-15
2-16 图示结构,已知L1 = 2L2 = 100 cm, 2A1 =A2 = 2 cm2, E = 200 GPa,α = 125×10−7 1/℃。
试求温度升高ΔT = 40℃时杆内最大应力。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0058_0005.jpg?sign=1734396067-PSs21iRoNWamTeJUagVy2x49VY5cXauH-0-97a6625abdfcafc7ae4623dd09f9f440)
习题图2-16
2-17 图示杆1、杆2和杆3与两刚性构件连接,杆1、杆2为材料相同的圆截面杆,d =10 mm,E = 200 GPa,L = 200 mm;杆3横截面面积A3 = 600 mm2,E3 = 100 GPa,由于制造误差,其长度为L + ΔL,ΔL = 0.2 mm。试求装配后各杆的应力。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0058_0006.jpg?sign=1734396067-ZSk6gEZj4g0SkQFz3sjLVhCs81J6RPYe-0-a8dc08f60de1a76013f868ddef361623)
习题图2-17
2-18 图示直径为d=8 mm的两根钢杆1、2并联。两根钢杆受力F=20 kN。钢杆的设计长度L=3m。[σ]=280MPa,E=200 GPa。加工时杆1比设计长度长了1.5 mm。计算两钢杆的应力和允许的最大误差。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0058_0007.jpg?sign=1734396067-rPiDQ7QAEJTYQ19ShFsfssQ8LAmlicY1-0-658a721d81d7a3077f184b28a252a524)
习题图2-18
2-19 剪刀如图所示,a = 30 mm,b = 150 mm,销钉C的直径d1 = 5 mm,当用力F = 200 N剪直径d2 = 5 mm的铜丝A时,试求铜丝及销钉剪切面上的切应力。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0058_0008.jpg?sign=1734396067-2i0b3uXTFzshQSXihi7MyDOm3UxDsCDW-0-5d432adfdcc39091ec75e010a19a2999)
习题图2-19
2-20 已知δ=2 mm,b=15 mm,d=4 mm, [τ]=100MPa,[σbs]=300MPa,[σ]=160MPa。试求许用载荷[F]。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0059_0001.jpg?sign=1734396067-KOGlbOOKbgzu3xUFi2ZiC1aXng8pWzwJ-0-ff942923487790ab5c4f55ee9a042fb3)
习题图2-20
2-21 已知F=45 kN,δ=10 mm,b=250 mm, h=100mm,l=100mm;顺木纹方向,许用应力分别为[τ]=1 MPa,[σbs]=10 MPa,[σ]=6MPa,试校核杆的强度。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0059_0002.jpg?sign=1734396067-n5ceN7qy90sZ9zch39IScAntlTgct6Kh-0-8f263a586f7bc6662c00d7fecbf79215)
习题图2-21
2-22 图示装置中键长L = 25 mm,其材料许用切应力[τ]=100 MPa,许用挤压应力[σbs]=220MPa,试求作用于手柄上的许用载荷[F]。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0059_0003.jpg?sign=1734396067-NuCn8ZPjPAma93AdiCG2Yf8mCAE9CoTM-0-854936396beb5be10ed812fe89d215bb)
习题图2-22
2-23 如图所示的杆,若材料的许用应力[τ] =100MPa,许用挤压应力[σbs]=320MPa,许用应力[σ]=160MPa,试求杆的许用载荷[F]。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0059_0004.jpg?sign=1734396067-wKvya3dlCcgnq4pS4XQGjpQznzekXsEu-0-edf9e6ddadccb73f46602a5b1c8c8493)
习题图2-23
提高题
2-24 图示结构,杆1、杆2和杆3的许用应力分别为[σ]1= 80 MPa,[σ]2= 60 MPa,[σ]3=120 MPa,弹性模量分别为E1 = 160 GPa, E2 = 100 GPa,E3 = 200 GPa,若F = 80 kN, A1 = A2 = 2A3,试确定各杆横截面面积。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0059_0005.jpg?sign=1734396067-r0wTmmhK8S4os3yWgFT3erYlsN7ML9CX-0-8a4cd515dd277752ec575c822df90cca)
习题图2-24
2-25 图示水平刚性梁AB,端A铰接,端B与杆 2 连接,杆 1 与刚性梁间的缝隙为0.06 mm,若杆 1 的截面面积为 A1 =2500 mm2,弹性模量E1 = 200 GPa,材料的许用应力[σ]1 = 100 MPa。杆2的截面面积为A2 = 1000 mm2,弹性模量E2 = 70 GPa,材料的许用应力[σ]2 = 120 MPa。试求当杆1与刚性梁在点D装配后结构的许用外载荷[F]。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0059_0006.jpg?sign=1734396067-IJZ1Aa9MfTDEjW98MRw0gBJWiIuRMkHg-0-c9424ebc02c789c98597a78a520f057b)
习题图2-25
2-26 正方形刚性板ABCD,由铰A和BF、DE两杆支撑。两杆抗拉(压)刚度为 EA, DE=2BF=2m,力F=6 kN。试计算两杆的内力。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0060_0001.jpg?sign=1734396067-jvPkW784CKLS0K6yYBExoJbwOBuflkwo-0-b0f56dc900ce82ec24be883d6a61ea3f)
习题图2-26
2-27 试利用能量法求解习题2-13。
2-28 图示简单桁架的杆件均由相同的钢材制成, E = 200 GPa,横截面面积均为300 mm2。若F = 5 kN,试求C点的水平和铅垂位移。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0060_0002.jpg?sign=1734396067-n1I5DOl2ACJLA0TwL2bz9SptlUyCMRrh-0-2f792e2d1fac620ea540bc789f57012b)
习题图2-28
2-29 如图所示,厚度t = 6 mm的两块钢板用3个铆钉连接,已知F=50kN,材料的[τ]=100MPa,[σbs]=280MPa,试确定铆钉直径d。若用d = 12 mm的铆钉,问需要几个?
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0060_0003.jpg?sign=1734396067-3ZFEKxO3V6FHRD0JDJe47s7efrZprNkv-0-b442b9a933307a4683eb4a6bdba84133)
习题图2-29
2-30 正方形截面的混凝土柱,其横截面边长为200 mm,其基底为边长a = 1 m的正方形混凝土板。柱承受轴向压力F = 100 kN,如图所示。假设地基对混凝土板的支反力为均匀分布,混凝土的许用切应力为[τ]=1.5MPa,试求混凝土板所需的最小厚度δ。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0060_0004.jpg?sign=1734396067-nMSAchePe91AMjZEMVSqi0TJnit1qfx3-0-32b6ee22d58c787d8409aff8cb8351f1)
习题图2-30
研究性题
2-31 图示BC、BD两杆原为水平位置,在力F的作用下两杆变形,点B的位移为Δ。若两杆的抗拉刚度同为 EA,试求Δ与F的关系。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0060_0005.jpg?sign=1734396067-5BnlNihRt0MTISNnNxi0uQtMSt1Ea6gt-0-e6fca6e8b426daad2b0ce9bd9aa630c7)
习题图2-31
2-32 图示桁架受力F1=F2=F的作用,试求节点D和E的垂直位移。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0060_0006.jpg?sign=1734396067-RpV63lL53He0YO30ozaKJIJcXnZMZCU7-0-a49cb162e14d7b35f8f9a45a588e2dfe)
习题图2-32
2-33 如图所示由两种材料完好粘接组合而成的复合杆件,横截面面积为A,两种材料弹性模量分别为E1和E2,各占一半体积,关于轴线对称分布。试分别就(a)、(b)两种情况:
(1) 绘制作用力 F 与复合杆件伸长量Δl之间的变化曲线;
(2) 杆件的总应变能和两种材料中分别存储的应变能;
(3) 若将复合杆等效为均匀材料,则等效刚度分别为多少?
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0061_0001.jpg?sign=1734396067-amAtf9lU08gomZirYOGv39gCJkIM85qb-0-7bd85194025cfedc206746ba584520c1)
习题图2-33
2-34 如习题图 2-33 所示由两种材料完好粘接组合而成的复合杆件,横截面面积为A,两种材料各占一半体积,关于轴线对称分布,其中材料 1是弹性模量为E1的弹性材料,材料2是如习题图2-34所示的弹塑性材料。试:(1) 试绘制作用力F与复合杆件伸长量Δl之间的变化曲线;(2) 作用力F达到某值F0,使材料2中的应力超过屈服极限,然后完全卸载,求杆的残余变形。
![](https://epubservercos.yuewen.com/C638BB/3590273103456001/epubprivate/OEBPS/Images/figure_0061_0002.jpg?sign=1734396067-ya4sSTrW44JDX4HGAIICWg02NWWw7ocU-0-ef06b1c2c4a39cc171d219090e8f1de0)
习题图2-34