2007年同济大学博士入学考试弹性力学 (2)
弹性力学试卷2007(答案)
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②
④
六、试用最小势能原理推导平面应力问题的位移形式的平衡方程和相应的静力边界 条件。 （设该平面的厚度为 1 单位，不考虑体力） （15 分） 解：在平面应力状态中， 0 ，应力应变关系为： z zy zx
E E E x , , x y y y x xy 2 2 1 1 2 1 xy
q
解： （1）应力函数需满足双调和方程，
2 2 1 1 U 0 2 2 2 r r r r 2
O
②
m n
x
经验证满足此方程。
（2）求应力：
2
y
图2
1 U 1 U 2 2C sincos sin2 tg r 2 r r r 2 U 2C sin cos cos 2 tg 2 r 1 U 2 2 C 1 sin cos 2 sin cos tg r r r
成绩 姓名 六 七 八
专业 二 三 四
班级 五
2 2 2 4. x＝K(x +y )， y＝Ky ， z＝0， xy＝2Kxy， yz＝0 ， zx＝0 。K 是不为零的已知 常数，这一组应变分量不可能存在。( × )
5. 理想弹性体中主应力方向和主应变方向相重合。(
× ) × )
6. 在所有静力可能的应力中，真实的应力使总势能取最小值。( 每题②分 二、填空题（10 分） 1.已知应变分量为：
y
xy
h y 2 h y 2
0 q ④
q l
h
x
次要边界：
y
同济大学2007年博士生入学考试试题 结构动力学
同济大学2007年博士生入学考试结构动力学
一、简答题（每题4分，共20分）
1. 结构自由度
2. 达朗贝尔原理
3. 用半幅值法推导
0.11
m
ς=。
4. 比较周期大小，三栋楼房A、B、C，每栋八层，其中在A座第四层和C座第八层分别放置相同质量，由大到小排列T A，T B，T C。
5. 写出单自由度系统欠阻尼的响应表达式。
二、计算题
1. 求自振周期。
图1
2. 悬臂梁，M1和M2由电磁吸引，去除电磁力使M2下落，求M1的振幅。
图2
3. 两层框架结构如图3所示，已知m1=m2=1kg，K1=2000，K2=4000，阻尼都为0.05。
1）求所有振型及自振频率；
2）求线性阻尼；
3）广义质量、广义刚度、广义阻尼；
4）用振型叠加法求稳态响应。
图3
4. 如图4所示，位移为3.038时自由释放，t=0.64s时再次达到最大，此时振幅为2.1844，质量M=90800kg，求刚度、阻尼和阻尼比。
图4
5. 求液体黏度系数u。
板重W，面积A。
在空气中自由振动，不计空气阻力，测得振幅为T1；放入液体中后，测得振幅T2，已知液体阻力F=πuAv（v为相对速度），求u。
图6
6. 推导悬臂梁横向振动的正交性。
（10分）
7. 求简支梁的振动频率。
（5分）。
同济大学弹塑性力学试卷及习题解答.
弹塑性力学试卷及习题解答弹塑性力学试卷配套教材《弹性与塑性力学》陈惠发1．是非题（认为该题正确，在括号中打√；该题错误，在括号中打×。
）（每小题 2 分）（1）物体内某点应变为0 值，则该点的位移也必为0 值。
（2）可用矩阵描述的物理量，均可采用张量形式表述。
3）因张量的分量是随坐标系的变化而变化，故张量本身也应随坐标系变化。
（）4）弹性的应力和应变张量两者的主方向是一致性，与材料无关的。
（）5）对于常体力平面问题，若应力函数x,y 满足双调和方程 2 20，那么，由x,y 确定的应力分量必然满足平衡微分方程。
（）（6）若某材料在弹性阶段呈各向同性，故其弹塑性状态势必也呈各向同性。
（）（7）Drucker 假设适合于任何性质的材料。
（）（8）应变协调方程的几何意义是：物体在变形前是连续的，变形后也是连续的。
（）（9）对于任何材料，塑性应变增量均沿着当前加载面的法线方向。
（）（10）塑性应变增量的主方向与应力增量的主方向不重合。
P107；226 （）2．填空题（在每题的横线上填写必要的词语，以使该题句意完整。
）（每小题 2 分）（1）设x,y a1x a2x y a3y ，当a1,a2,a3满足_________________________________ 关系时x,y 能作为应力函数。
（2）弹塑性力学是研究固体受外界因素作用而产生的______________________ 的一门学科。
（3）导致后继屈曲面出现平移及扩大的主要原因是材料_______________________ 。
（4）π 平面上的一点对应于应力的失量的 _____________________ 。
P65（5）随动强化后继屈服面的主要特征为：__________________________________________ 。
（6）主应力轴和主应变轴总是重合的材料为_______________________ 。
《弹性力学》试题参考答案(参考题)
《弹性力学》试题参考答案（答题时间：100分钟）一、填空题（每小题4分）1．最小势能原理等价于弹性力学基本方程中： 平衡微分方程 ， 应力边界条件 。
2．一组可能的应力分量应满足： 平衡微分方程 ，相容方程（变形协调条件） 。
3．等截面直杆扭转问题中， M dxdy D=⎰⎰2ϕ的物理意义是 杆端截面上剪应力对转轴的矩等于杆截面内的扭矩M 。
4．平面问题的应力函数解法中，Airy 应力函数ϕ在边界上值的物理意义为 边界上某一点（基准点）到任一点外力的矩 。
5．弹性力学平衡微分方程、几何方程的张量表示为：0,=+i j ij X σ ，)(21,,i j j i ij u u +=ε。
二、简述题（每小题6分）1．试简述力学中的圣维南原理，并说明它在弹性力学分析中的作用。
圣维南原理：如果物体的一小部分边界上的面力变换为分布不同但静力等效的面力（主矢与主矩相同），则近处的应力分布将有显著的改变，但远处的应力所受影响可以忽略不计。
作用：（1）将次要边界上复杂的面力（集中力、集中力偶等）作分布的面力代替。
（2）将次要的位移边界条件转化为应力边界条件处理。
2．图示两楔形体，试分别用直角坐标和极坐标写出其应力函数ϕ的分离变量形式。
题二（2）图（a ）⎩⎨⎧=++= )(),(),(222θθϕϕf r r cy bxy ax y x （b ）⎩⎨⎧=+++=)(),(),(33223θθϕϕf r r dy cxy y bx ax y x3．图示矩形弹性薄板，沿对角线方向作用一对拉力P ，板的几何尺寸如图，材料的弹性模量E 、泊松比 μ 已知。
试求薄板面积的改变量S ∆。
题二（3）图设当各边界受均布压力q 时，两力作用点的相对位移为l ∆。
由q E)1(1με-=得，)1(2222με-+=+=∆Eb a q b a l设板在力P 作用下的面积改变为S ∆，由功的互等定理有：l P S q ∆⋅=∆⋅将l ∆代入得：221b a P ES +-=∆μ显然，S ∆与板的形状无关，仅与E 、μ、l 有关。
2007年同济大学博士研究生入学考试试题
同济大学2007年博士研究生入学考试试题编号：101 考试科目：英语答题要求：答案一律写在答题纸上。
1． Vocabulary(10%)Directions：For each of the following sentences there are four choices. Choose the best one to complete the sentence. Mark your choices on the answer sheet.1．The man had a good disguise, but as soon as he spoke he ______himself．A．exposed B．revealed C．betrayed D．disclosed2. The leaders of the two countries feel it desirable to ______funds from armaments to health and education．A．derive B．deprive C. dispatch D．divert3．Democratic government is a phrase that is notoriously hard to ________.A．credit B．defy C．modify D．define4. I reject absolutely the _____that privatization is now inevitable in our industry．A. perception B．notion C．impression D．concept5. With the economy of the county going strong，the ______mood is one of optimism．A．presiding B，circulating C．floating D. prevailing6. The panel will consider whether or not Mr．Wilson has been ________serious professional misconduct.A．wary of B．guilty of C．confronted with D．reduced to7. It is a ____of our company to give refunds if goods are faulty.A．policy B．discipline C．decision D．determination8. He had always been _______ the way Ruth looked, and had never once paid her a complimentA．unaware of B．cautious about C．oblivious to D. subject to9．Since a circle has no beginning or end，the wedding ring is a symbol of _______love．A．constant B．infinite C．prolonged D．eternal10．Dick，who had failed the math test，was sitting．on a bench in the corner, ______over his disappointment．A．brooding B．mediating C．apologizing D．complaining11. Last week the seamen＇s strike led to the _______closure of the whole vast dock area．A．actual B．virtual C．factual D．local12．In the garden bees moved from _________flowers to purple ones．A．colorful B．prosperous C．scarlet D．brilliant13．Professor Smith has already retired but his teachings still _____a strong influence on his students.A．perform B．exhibit C．exert D. execute14．When they asked him about it，he said it was no ______ of theirs and wouldn＇t tell them anything.A．concern B．relation C. connection D．relevance15．The hunter knows quite well that wild animals go seeking the their _______ in the jungle．afterdark．A．victim B．favorite C．prey D．sacrifice16．I have absolutely no ______of ever meeting him before．A．reminiscence B．reminder C．recall D．recollection17. They agreed to take their disputes before the committee and ______ by its decisions.abideobserve C.D.precedeA.standB.18. Every weekend when I came back from school, Mother prepared meals ______ enough for a Sahara-bound camel and made me eat them up.A. adequateB. deliciousC. proficientD. substantial19. As the fat man sat down, the folding chair ______ under him, with a loud noise of tearing canvas.sustainedA.distorted D.yielded B.collapsed C.20. Modern artists often need financial support but they have difficulty in finding wealthy ______.millionaires D.patrons C.volunteersA.admirers B.II. Reading Comprehension (50%)Direction: For each of the following passages there are five questions or unfinished is followed by five statements. For each of them there are four choices, choose the best one to answer the question or complete the statement.Passage 1In managing information resources, the medium may be the key to an effective system. The medium is a vehicle, a tool, or a container for holding information; the information itself is the thing of value.There popular categories of information media are paper, film, and electronic system. The media choice must not be viewed as a choice among these three, however, it must be viewed as an opportunity to select from a multitude of media possibilities in combinations, that build effective systems. In many instances the person responsible for information-resource management is not the person who determines the medium in which information will be created. In such a case, the manager of a firm＇s information resources faces a challenge in making a significant contribution to the organization＇s objectives.For effective management of information resources, media conversion may be necessary. examples include keying or scanning paper documents to convert them to electronic media. Other processes convert electronic media from one format to another. For example, disk files created on one system may not be compatible with another system. Various hardware and software combinations can be used to convert files to formats that equipment will accept. For information generated within organizations, this necessity of making systems compatible may be eliminated by cooperative planning. However, very little control can be exercised over the media used to generate information that comes to your organization from the outside.The medium for information may be selected to satisfy. a need that excite when information is created and communicated. For example, a paper record may be created because of its portability and because no special equipment is necessary for later references to that information; electronic transmission may be selected because it is the fastest means of communicating information. A firm may use electronic mail because a network already exists for on-line computercommunication. The additional application may cost less than postage to mail paper memos.21. Which of the following can best sum up the passage?A. Media Selection in Managing Information Resources.B. The Importance and Necessity of Media ConversionC. Three Categories of Information Media.D. Various Means of Communicating Information22. The first paragraph aims telling the reader ______.A. the importance of information resources managementB. the relationship between the medium and informationC. the great variety of media for holding informationD. the numerous resources of information23. According to the author, ______.A. paper is the best storage deviceB. people have three choices in selecting information mediaC. it is better to let the person responsible for information-resource management determine the mediumD. the manager should build an effective system by selecting a good combination of different media24. For effective management of information resources, the manager should ______.A. convert all paper documents to electronic mediaB. make media conversion when necessaryC. control the media used to generate information both inside and outside his organizationD. use one format in processing information25. The main idea of the last paragraph is ______.A. paper record is the most convenient medium for later referenceB. electronic mail costs less than postage to mail paper memosC. different media for information may be selected for different purposesD. by using different media, a firm can create various in formation for its objectives Passage 2＂digital, digital, digital＂ continues to grow in volume worldwide. Digital ofchantThecameras, digital video camcorders, video CD players, DVD, cellular phones, and a host of computer peripherals are moving the trend along at a breathtaking rate. For the average person, it may seem like a remote and puzzling phenomenon meant only for the technologically adept.Virtually every aspect of our lives could be affected by the digital revolution. Here is a hypothetical scenario to show the possibilities: A real estate agent in Seattle uses a digital still camera to take some pictures of a house she＇s trying to sell., She transfers them to her computer, digitally retouches and enhances them, and posts them on her company＇s Internet Web site. In Singapore, a buyer sees the pictures and asks via electronic mail for more information. The agent replies via e-mail and attaches the text and a digital video clip to her message. Later the buyer flies to Seattle, inspects the property, and seals the deal.One of the biggest marketing surprises of the current age is the digital still camera. Once prohibitively expensive, these cameras have radically dropped in price while gaining in resolutionand other features. Although they often resemble traditional cameras, they don＇t use film. Instead, they store images on either a small removable memory card or on the memory chip inside the camera.The beauty of digital photography is that while you＇ll spend relatively more for a digital camera, you＇ll save a lot on film processing costs, because there aren＇t any. You can also discard digital pictures and keep shooting. Better yet, you can use software to enhance or alter the image. In quality, the images consumer-level digital cameras produce do not compare to ones you＇d get from a 35mm camera. For the most part, though, digital photos are meant to be viewed on a computer monitor, and so their resolution is more than acceptable. In a world where the speed at which you distribute information often means the difference between success and failure, and immediacy supersedes quality in importance, may people are finding a use for digital camera.26. From the first paragraph we know the average person thinks ____.A. the digital age is far awayB. digital is too complicated for himC. digital age is dreadfulD. both a and b27. In the second paragraph the writer used an example to show that _____.A. any real estate agent should use digital still cameraB. digital still camera should be used together with Internet WebC. digital age is gaining momentumD. house dealing can be made easier28. _____ could be the best summary of the last two paragraphs.A. Digital still camera and traditional cameraB. Pros and cons of digital still cameraC. Capturing the world digitallyD. The way to use digital still camera29. All the following are the advantages of digital still camera EXCEPT ______.A. no need of filmB. easy to carryC. the image can be changedD. high resolution30. Which of the following best summarizes the passage?A. New Digital Age of InteractivityB. Digital Still CameraC. One Application of ComputerD. Goodbye Analog, Hello DigitalPassage 3In the days immediately following hurricane Andrews deadly visit to South Florida, Allstate Insurance hastily dispatched more than 2,000 extra clang adjusters to the devastated area to assist the 200 stationed there. Many of the reserves rived in convoys of motor homes. Others flew in from as far away as Alaska and California. Since the storm had knocked out telephone lines, Allstate rushed to set up its own communications system. Allstate expects to pay out $1.2 billion to cover more than 121,000 damage claims as a result of Andrew.All told, U.S, property and casualty insurers have been hit with more than $8 billion in Andrew-related claims, making the hurricane the most costly single calamity to strike the industry since the San Francisco earthquake and fire in 1906 ( cost: $6 billion, after inflation). With claims continuing to pour in, Andrew threatens to take a painful toll on the already battered property-casualty insurance industry and its 100 million policy-holders. The final bill, analysts predict, is likely to top $10 billion. While most well-capitalized insurers are expected to weather the storm, less anchored firms are in danger of being blown away, leaving U.S. consumers stuck with the tab. Says Sean Mooney, senior researcher at the Insurance Information Institute: ＂It will take years before the industry digs itself out from the wreckage left by Andrew. Some (companies) will be buried by it?Hurricane Andrew is the latest in a string of mishaps to plague the American insurance industry this year. In April an over flowing Chicago River flooded the city＇s downtown district, costing insurers $300million in claims. A month later, Los Angles was rocked by the worst civilian riot in the U.S. since the Civil War. The insurance toll: $1 billion. Then came a series of major hailstorms in Texas. Florida and Kansas. They cost insurers a combined $700 million. And two weeks after Andrew, another lethal hurricane, Iniki, smashed into Hawaii, causing $ 1.4 billion in damages. In all, property and casualty insurers have paid out a record $13 billion in claims so far this year, far surpassing the previous high of $7.6 billion in 1989, the year of Hurricane Hugo and California＇s Bay Area earthquake. Just as in that year, when those catastrophes were followed by substantial increases in insurance premiums, insurers are already lobbying for rate relief.31. According to the passage. ＂Allstate Insurance＂most likely refers to ____.A. one of the property and casualty insurers in the U.S.B. the only insurance company responsible for the damage claims by AndrewC. the insurance industry as a wholeD. the biggest insurance company in the U.S.32. As is stated in the second paragraph, the result of hurricane Andrew is likely to _______.A. lead to inflation throughout the U.S.B. make the largest insurers suffer the mostC. put the industry in South Florida out of actionD. cause insurers with insufficient funds to go bankrupt33. Using context clues, we may infer that ＂stuck with the tab＂most probably means _____.＂caught in the hurricane＂A.＂exposed to natural disasters＂B.＂trapped in financial difficulties＂C.＂extremely vulnerable to further damages＂D.35. The main purpose of the passage is to _____.A. show the severe damages and heavy losses caused by Hurricane AndrewB. suggest that U.S. insurers are virtually unable to cover the damage claims any moreC. tell about the difficult situation faced by the insurers throughout the U.S.D. prove that disasters tend to cause ever worsening devastation as time goes on.Passage 4Steve Courtney wrote historical novels. Not, he was quick to explain, over colorful love stories of the kind that made so much money for so many women writer, but novels set, andcorrectly set, in historical periods. Whatever difference he saw in his own books, his readers didnot seem to notice it, and his readers were nearly all women. He had studied at university, but hehad not been a particularly good student, and he had never afterwards let any academic knowledgehe had gained interfere with his writing.Helen, his wife, who did not have a very high opinion of her husband's ability as a novelist,had been careful to say when she married him that she was not historically minded.Above all, Helen was doubtful whether her relationship with Steve would work at all in thevillage of Stretton, to which they had just moved. It was Steve who had wanted to move to thecountry, and she had been glad of the change, in principle, whatever doubts she was now havingabout Stretton as a choice. But she wondered whether Steve would not, before very long, wantto live in London again, and what she would do if he did. The Stretton house was not a weekendcottage. They had moved into it and given up the London flat altogether, partly at least, she suspected, because that she Steve's idea of what a successful author ought to do. However, she thought he was not going to feel like a successful author half as much in Stretton ashe had in London. On the other had, she supposed he might just start dashing up to London forthe day to see his agent or have lunch with his publisher, leaving her behind in Stretton, andshe thought on the whole she would like that.36. What was Steve＇s attitude towards women who wrote love stories?A. He would have liked to earn as much money as they did.B. He was afraid of being compared unfavorably with themC. He did not think he could write about the same subjectsD. He had a low opinion of the kind of books they wrote37. What did Helen have to be careful to hide?A. Her lack of interest in historyB. Her low opinion of her husband＇s writingC. Her dislike of her husband＇s writingD. Her inability to husband＇s admirers.38. What were Helen＇s feelings about the move from London to Stretton?A. She wanted to remain in the countryB. She had been unwilling to leave LondonC. She thought it was time to return to undon39. Helen thought Steve might not be content in Stretton because ______.A. he would not be able to write so well in the countryB. he would not feel so important in StrettonC. his relationship with Helen was changingD. he would not be lonely without all his London friends40. The passage as a whole suggests that Steve＇s novels were _____.A. popular but unimportantB. serious works of literatureC. admired for their historical truthD. written with women readers in mindPassage 5The sixties have been misunderstood. It was not a radical decade, as the term radical is commonly used in connection with those years. It was not a decade of the left ascendant. Rampant, perhaps but not ascendant. Rather, the decade was radical radicalizing, which, subsequent decades have shown, is different. Politically, the sixties invigorated the right more than the left. But of course politics is not everything. In fact, three decades down the road, the nation＇s political discourse may be driven by conservatives, but they, although by many measures triumphant, seem aggrieved because politics seems peripheral to, and largely impotent against, cultural forces and institutions permeated with what conservatives consider the sixties sensibilities.Treating a decade as a discrete entity obviously makes the assumption that history during that decade had an obliging tidiness, opening with a decisive and tone-setting episode and closing with a suitably climatic event. History rarely accommodates that assumption. Such a treatment of a decade also makes the equally dubious assumption that the decade in question had a clearly dominant tone or profile. So the 1920s was the decade of jazz., flappers, the birth of the sports celebrity (Babe Ruth, Red Grange, Jack Dempsey), the Lost Generation, Sacco and Vanzetti and ...Warren Harding, Calvin Coolidge, and Herbert Hoover.Let us stipulate this, then: A decade, even one as intensely felt at the time and as hotly debated afterward as the sixties was and is, can come to seem, when recollected in tranquility, quite unlike the decade as it was felt at the time, and unlike the decade as it is portrayed by people with an emotional or political investment in portraying it a particular way.It is arguable that we should think of the sixties as beginning in November 1963 and ending in October 1973. That is, the years we connect with the tumultuousness associated with the phrase ＂the sixties＂ began with the assassination of a president and ended with the Yom Kippur War and the energy crisis. The assassination shattered (or at least many people say it did ) the nation＇s sunny postwar disposition; it supposedly ＂ended American innocence.＂ It is unclear how innocent was this nation that had been made possible by Puritans, had been founded by such innocents as Benjamin Franklin, John Adams, and James Madison, had been born in the bloodshed of what actually was not only the American Revolution but also America＇s first civil war, had been preserved by the worst civil war the world had until then seen...you get the picture. The sixties as a decade of ＂lost innocence＂? Please. The 1973 oil embargo, which produced a sense of national vulnerability and pervasive limits, did seem to bring down a curtain on something. But on what?41. What does the passage mainly discuss?A. A radical period in American historyB. Different events in the sixtiesC. The true picture of the sixtiesD. Real radicals in the sixties42. Which of the following features describes the sixties according to the passage?RadicalA.AscendantB.WithdrawingC.RampantD.43. Which of the following can be inferred from the passage?A. The sixties should not be judged separatelyB. It is impossible to make assumptions about any decadeC. People tend to misunderstand some events in historyD. History presents many confusing phenomena to people44. It can be inferred from the passage that people regard the sixties asdangerousA.peacefulB.chaoticC.unbearableD.45. It can be most safely concluded from the passage that the 1973 oil embargo _____.A.America＇s weaknessreflectedAmerica＇s great power in the worldB.showeddidn＇t affect America at allC.D.America＇s position in the worldchangedIII． Translation (20%)Direction: Translate the following passage in to Chinese. Write your translation on the answer sheet.It is not enough to know what is right and what is wrong. One must also be able to apply this knowledge to actual, concrete situations. For that, the virtue of prudence is essential.St Thomas Aquinas referred to prudence as the ＂rudder virtue＂ the one that ＂steers＂the others. Without it, we are like someone adrift in a boat, tossedin this direction or that by the wind, the waves, and the current. Unfortunately, the virtue of prudence has too often been confused with caution. Thus, the ＂prudent＂person is one who never ＂rocks the boat＂and is especially careful to aroid is the wisest and most moral course of action to take.The word that actually come closest to fulletining as a synonym for prudence is ＂discernment＂. The prudent person is one who can ＂size Up a situation and decide. What is the, what is the wisest and most moral course of action to take.prudencedoesn＇t answer the question, ＂What is the right thing in principle to Accordingly,do? Rather, ＂What is the right thing for me (or for us) to do in this situation?＂The prudent person fully examines a situation and seeks advice from others. (Reaching out to others for counsel before acting is one of the classic marks of a prudent person.) A judgment is made in the light of this examination and advice, and then a decision is made.IV. Writing (20%)Directions: Read the following sentences carefully and write a composition of 250-300words according to the requirement. Write your composition on the answer sheet.Do you agree or disagree with the following statement? Face-to-face communication is better than other types of communication, such as letters, e-mail, or telephone calls. Use specific reasons and details to support your answer.Key to Paper AI. Vocabulary (10%)1-10: C D D B D B A C D A11-20: B C C A C D C D B BII. Reading comprehension (50%)21-30: A B D B C D C C B A31-40: A D C A C D B D B A41-45: C D A C AIII. Translation (20%)只知道什么是对，什么是错，这还不够，还必须能把这一知识运用于实际的具体情况．为此，审慎这一美德便是必不可少的。
弹性力学_同济大学
变形前p x, y,变形后 pxu,yv.
思考题
1. 试画出正负 y 面上正的应力和正的面力 的方向。
2. 在d x d y 1的六面体上，试问x面和y面 上切应力的合力是否相等？
第一章 绪 论
研究方法
§1－3 弹性力学中基本假定
弹性力学的研究方法，在体积V 内： 由微分体的平衡条件，建立平衡微分方程；
正应变 x , y，以伸长为正。
切应变 xy, 以直角减小为正,用弧度表示。
第二节 弹性力学中的几个基本概念
正的正应力对应于正的线应变, 正的切应力对应于正的切应变。
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第二节 弹性力学中的几个基本概念
位移
位移 -- 一点位置的移动，用 u, v表示,
第一节 弹性力学的内容 第二节 弹性力学中的几个基本概念 第三节 弹性力学中的基本假定
第一章 绪 论
定义
§1-1 弹性力学的内容
弹性力学 --研究弹性体由于受外力、边 界约束或温度改变等原因而发生的应力、形 变和位移。
研究弹性体的力学，有材料力学、结构 力学、弹性力学。它们的研究对象分别如下：
第一节 弹性力学的内容
（表示） σ x-- x 面上沿 x向正应力， xy-- x 面上沿 y向切应力。
（符号）应力成对出现，坐标面上的应 力以正面正向，负面负向为正。
第二节 弹性力学中的几个基本概念
例：正的应力
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第二节 弹性力学中的几个基本概念
同济大学2007年硕士生入学考试 808材料力学与结构力学试卷及解答
2 σ 2 + 3τ= 2 = = 36.462 + 3 ×14.25 44.03MPa < [σ ] 200MPa
八、写出图示结构的位移法方程，并求方程的系数和自由项。 （15 分）
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同济大学 2007 年硕士生入学考试材料力学与结构力学试卷及解答
满分分值:150 答题要求：1.答题一律做在答题纸上，做在试卷上无效。 2.考试时间 180 分钟。 一、图示由铜套管和钢圆柱组成的构件置于刚性平面和刚性垫块之间，已知套管和 钢 柱 的 横 截 面 分 别 为 Ac = 6.5cm 2 和 As = 8.0cm 2 ， 其 材 料 弹 性 模 量 分 别 为
同济大学弹性力学第二章-修改20140226
2.1 体力和面力 2.2 应力和一点的应力状态 2.3 与坐标倾斜的微分面上的应力 2.4 平衡微分方程 应力边界条件 2.5 转轴时应力分量的变换 2.6 主应力 应力张量不变量 2.7 最大切应力
第二章 应力状态理论
弹性力学研究的都是超静定问题，故而必 须从静力学、几何学和物理学三个方面一起考 虑。
2.4 平衡微分方程 应力边界条件
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同济大学博士生入学考试试题
本资料属个人考试后回忆内容，翔实可靠，但错误在所难免，望广大学友指正土木工程学院结构工程专业攻读博士学位入学考试专业课试题及若干注意事项（结构动力学和地震工程学版）结构工程是这样一种艺术；使用材料这些材料属性只能估算建立真实的结构这些真实的结构只能近似分析来承受外力这些力不能准确得知以满足我们对公众安全职责的要求爱德华·L·威尔逊目录土木工程学院结构工程专业攻读博士学位入学考试专业课试题及若干注意事项 (1)目 录 (1)同济大学及其土木工程学院简介 (2)若干注意事项 (4)同济大学2005年博士生入学考试《地震工程学》真题 (6)1.名词解释 (6)2.填空题 (6)3.简答题 (6)4.计算题 (6)同济大学2003年博士生入学考试《地震工程学》真题 (7)1.名词解释 (7)2.填空 (7)3.简答题 (7)4.讨论分析 (7)同济大学2007年博士生入学考试《地震工程学》真题 (8)1.名词解释（每题4分，共20分） (8)2.填空（每空3分，共24分） (8)3.简答（共44分） (8)4.分析讨论（共12分） (8)同济大学2007年博士生入学考试《结构动力学》真题 (9)1.简答（每题4分，共20分） (9)2.计算题 (9)同济大学2003年博士生入学考试《结构动力学》真题 (12)1.名词解释（10分） (12)2.计算题 (12)同济大学2005年博士生入学考试《结构动力学》真题 (14)1.简答题 (14)2.计算题 (14)同济大学及其土木工程学院简介同济大学是教育部直属重点大学。
她创建于1907年，早期为德国医生在上海创办的德文医学堂，取名"同济"意蕴合作共济。
1912年增设工学堂，1923年被批准改名为大学，1927年正式定为国立同济大学。
抗战期间曾内迁经浙、赣、滇入川，1946年回迁上海并发展为以拥有理、工、医、文、法五大学院著称海内外的综合性大学。
同济大学07-08工程力学2试卷 (答B)
同济大学课程考核试卷（B 卷）答案：一、填空题（每小题5分，共20分） 1.答：5m/s 2 ；[4分] 与O z B 成60度角。
[5分] 2.答：43mRv；[2.5分] 1632m v 。
[5分]3．答：惯性力系主矢的大小为：me α，me ω2；[2.5分] 惯性力系的主矩的大小为：2)2(22αe R m +（图略）。
[5分] 4．答：2个；[1分] 2sin ϕPL - ；[5分]二、计算题（10分）解：以杆CD 上的D 点为动点，动系固连于杆AB 。
r e v v v+= [5分]L Lv v 4sin 22e ωθω=== （水平向右） [10分] 三、计算题（15分） 解：ω⨯=OA v A ，2nω⨯=OA a AC 点为AB 杆的速度瞬心s rad/25.11==ACv Aω（逆钟向） [4分] BC 杆的角速度12ωω=t n n t n AB AB A B B a a a a a ++=+22n ω⨯=BC a B ，21n ω⨯=AB a BA[7分]BC ：ϕϕϕsin cos cos t n n n BA BA A B a a a a ++-=得 2t cm/s 61.73=BA a2t1s rad/07.3==ABa BAα（顺钟向） [10分]四、计算题（10分） 解：设系统动量向下为正，其大小为22v g P v g P v g P p C B A --=)2(2 C B A P P P g v--= [10分] 解：对轮Ａ：m 1 a A = F S A －F A C ○1 [4] J A α A = M －F S A R ○2 [8] 对轮Ｃ：J C α C = ( F A C －F C D ) r ○3 [12] 对轮Ｄ、物Ｂ：m 3 a = F C D + F D E －m 3 g ○4 [16] 式中：F C D = F D E =21m 3 ( g + a ) ， a A = r α C = 2 a [18] 由○1、○2、○3、○4可得： a =Rm m m gRm M )26(23213++- F S A = R M －21a mF N A = m 1 g [20]六、计算题（10分）解：AC 杆和BC 杆均作“瞬时平移”。
同济大学弹塑性力学试卷及习题解答教学文案
弹塑性力学试卷及习题解答弹塑性力学试卷配套教材《弹性与塑性力学》陈惠发1．是非题（认为该题正确，在括号中打√；该题错误，在括号中打×。
）（每小题2分）（1）物体内某点应变为0值，则该点的位移也必为0值。
（） （2）可用矩阵描述的物理量，均可采用张量形式表述。
（ ） （3）因张量的分量是随坐标系的变化而变化，故张量本身也应随坐标系变化。
（ ） （4）弹性的应力和应变张量两者的主方向是一致性，与材料无关的。
（）（5）对于常体力平面问题，若应力函数()y x ,ϕ满足双调和方程022=∇∇ϕ，那么， 由()y x ,ϕ确定的应力分量必然满足平衡微分方程。
（） （6）若某材料在弹性阶段呈各向同性，故其弹塑性状态势必也呈各向同性。
（ ） （7）Drucker 假设适合于任何性质的材料。
（ ） （8）应变协调方程的几何意义是：物体在变形前是连续的，变形后也是连续的。
（ ） （9）对于任何材料，塑性应变增量均沿着当前加载面的法线方向。
（ ） （10）塑性应变增量的主方向与应力增量的主方向不重合。
P107；226 （ ）2．填空题（在每题的横线上填写必要的词语，以使该题句意完整。
）（每小题2分）（1）设()4322241,y a y x a x a y x ++=ϕ，当321,,a a a 满足_______________________关系时()y x ,ϕ能作为应力函数。
（2）弹塑性力学是研究固体受外界因素作用而产生的______________________的一门学科。
（3）导致后继屈曲面出现平移及扩大的主要原因是材料______________________。
（4）π平面上的一点对应于应力的失量的______________________。
P65 （5）随动强化后继屈服面的主要特征为：___________________________________________。
2007博士学位研究生入学考试试题答案
1、什么是软地层：
常规单极子声波测井测量的是在井壁地层中传播的滑行波，包括滑行纵波和滑行横波。
滑行波产生的条件是井中流体（泥浆）的声速小于地层介质的声速，这对于纵波来说不成问题，但横波速度相对较小，在有些地层条件下就可能不满足滑行波产生的条件（即可能地层横波速度小于井内流体声波速度），导致有些地层无法测量到滑行横波。
通常把这种地层中横波速度小于井内流体声速的地层称为软地层，其他的相应就称硬地层。
新一代的声波测井如DSI等，很重要的一个方面就是借助挠曲波获取各种地层的横波速度。
通常把地层横波速度小于井内流体波速的地层称为软地层，也就是说单极子声波测井在井能无法产生滑行横波。
同样，DSI（偶极子声波测井仪）也无法在井能产生滑行横波。
但DSI产生的剪切扰曲波在高频时其速度低于地层横波速度，低频时与横波速度相同。
所以可以用DSI的扰曲波提取地层的横波时差。
2、孔喉结构可以反映在核磁共振T2谱上。
在含流体的多孔砂岩中，T2的衰减主要取决于表面驰豫，而表面驰豫时间与比表面成反比，比表面又于孔径成反比，所以驰豫时间与孔径成正比，因此孔径越大，衰减越慢，T2越长，反之，孔径越小衰减越快，T2越短。
因此在注水开发中，地层孔喉结构的变化可以通过T2谱分布特征的改变而得以了解。
同济大学弹性力学期末试卷06-07
A卷
2006—2007 学年第 一 学期 课程名称：弹性力学 课号： 任课教师： 专业年级： 学号： 姓名： 考试（√）考查（ ） 考试（查）日期： 2007 年 1 月 22 日 出考卷教师签名：朱合华、许强、王君杰、李遇春、陈尧舜、邹祖军、赖永瑾、 蔡永昌 教学管理室主任签名：
A
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3P P 3 3P 1 xy cos sin (1 y), y 0, xy cos ( 2 y 2 1) 3 4a 4a 8a 16a 4a
将应力分量代入边界条件，可解得：
A q, B
所以应力分量解答为：
1 q cos 2
r q(cos cos ) q(cos 2 cos ) r q sin
(2) 解：由题可知，体力 X=0，Y=0，且为弹性力学平面应力问题。 1） 、本题所设应力函数满足双调和方程：
2 2
(√)
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确定的应力分量必然满足平衡微分方程。 果会有所差别。 （4）如果弹性体几何形状是轴对称时，就可以按轴对称问题进行求解。
（3）在求解弹性力学问题时，要谨慎选择逆解法和半逆解法，因为解的方式不同，解的结
（5）无论是对于单连通杆还是多连通杆，其载面扭矩均满足如下等式：
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最新同济【弹性力学试卷】期终考试A-本科
1、DIY手工艺市场状况分析 (4分)
边界条件为
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（2）当 ；
（二）DIY手工艺品的“热卖化”（3）当 ；
（4）当 。
（二）创业弱势分析这些条件给出如下方程组
(5分)
我们认为：创业是一个整合的过程，它需要合作、互助。大学生创业“独木难支”。在知识经济时代，事业的成功来自于合作，团队精神。创业更能培养了我们的团队精神。我们一个集体的智慧、力量一定能够展示我们当代大学生的耐心.勇气和坚强的毅力。能够努力克服自身的弱点，取得创业的成功。解出这一方程组后，得
(2)求最大和最小剪应力值。
2.已知应变张量:
试求:(1)主应变;(2)主应变方向;(3)应变不变量
三．计算题（共50分）
1、(13分)如图1所示为一等截面简支梁，抗弯刚度为 ，长为 ，受均布荷载 和集中力 作用，试采用瑞利——李兹(Rayleigh-Ritz)法或者伽辽金(Galerkin)法（二法只选一种）求挠度函数 。
13.若整个物体没有刚体位移，则物体内任意点处的微元体都没有刚体位移。（×）
14.出现最大剪应力的微平面和某两个应力主方向成45度角。（√）
15.对任意弹性体，应力主方向和应变主方向一致。（×）
二．分析题（共20分，每小题10分）
1．
(1)利用方程 ，即得
再利用 ，可得
(6分)
(2)由于应力张量 的剪应力分量全为零，因此3个正应力均为主应力。又由于最大主应力为 ，最小主应力为 ，因此最大最小剪应力分别为
和 ，它们所对应的微分面法矢量与 轴垂直，且与 轴和 轴的夹角（不计方向）均为45度。(4分)
2．
同济大学弹塑性力学试卷及习题解答(完整资料).doc
【最新整理，下载后即可编辑】弹塑性力学试卷及习题解答弹塑性力学试卷配套教材《弹性与塑性力学》陈惠发1．是非题（认为该题正确，在括号中打√；该题错误，在括号中打×。
）（每小题2分）（1）物体内某点应变为0值，则该点的位移也必为0值。
（ ）（2）可用矩阵描述的物理量，均可采用张量形式表述。
（ ）（3）因张量的分量是随坐标系的变化而变化，故张量本身也应随坐标系变化。
（ ）（4）弹性的应力和应变张量两者的主方向是一致性，与材料无关的。
（ ）（5）对于常体力平面问题，若应力函数()y x ,ϕ满足双调和方程022=∇∇ϕ，那么，由()y x ,ϕ确定的应力分量必然满足平衡微分方程。
（ ）（6）若某材料在弹性阶段呈各向同性，故其弹塑性状态势必也呈各向同性。
（ ）（7）Drucker 假设适合于任何性质的材料。
（ ）（8）应变协调方程的几何意义是：物体在变形前是连续的，变形后也是连续的。
（ ）（9）对于任何材料，塑性应变增量均沿着当前加载面的法线方向。
（ ） （10）塑性应变增量的主方向与应力增量的主方向不重合。
P107；226 （ ）2．填空题（在每题的横线上填写必要的词语，以使该题句意完整。
）（每小题2分）（1）设()4322241,y a y x a x a y x ++=ϕ，当321,,a a a 满足_______________________关系时()y x ,ϕ能作为应力函数。
（2）弹塑性力学是研究固体受外界因素作用而产生的______________________的一门学科。
（3）导致后继屈曲面出现平移及扩大的主要原因是材料______________________。
（4）π平面上的一点对应于应力的失量的______________________。
P65（5）随动强化后继屈服面的主要特征为：___________________________________________。
同济大学博士研究生入学考试结构动力学试题(2007年回忆)
结构动力学（不完全版）
简答：
1）什么是结构的自由度？
2）简述达朗贝尔原理。
3）a）一个八层的框架结构；b）在结构四层放一质量块；c）在结构顶层放一质量块；
比较a），b），c）三个结构的周期。
4）有阻尼体系的振幅衰减，在M个周期后，振幅降低到一半，求结构阻尼比。
计算：
1）求结构的阻尼比和阻尼系数；
2）求结构动能、势能和频率；
3）证明多自由度体系的模态正交性；
4）双自由度体系，在其中一个质量上作用简谐荷载，求结构频率、振型，阻尼矩阵，广义质量，广义刚度和广义阻尼；用振型叠加法计算结构位移。
（计算量大）
5）列出均匀弹性简支梁的运动方程。
注：考试内容主要参照考试大纲，以单自由度，双自由度为主，会求结构的频率，阻尼等基本参数，以及熟练掌握模态叠加法。
考博用弹性力学核心总结
空间位移分量表示弹性方程 按位移求解空间问题平衡微分方程 无限半空间采用按位移求解代入平衡微分方程 可导出积分得 代入弹性方程得 再根据边界条件求 基本方程：①平衡微分方程：
还有一种： 或者
②几何方程： ③物理方程：（按应力）
空间轴对称平衡微分方程：
把变成按应变问题
空间轴对称几何方程
空间轴对称物理方程： 按位移求解空间轴对称问题：弹性方程：力分量用形变表示： ，， 拉梅方程：
应力法（按应力求解的方法）：取应力分量为基本未知函数，从 基本方程和边界条件中消去位移和应变分量，导出用应力表示的相容方 程和边界条件。并由此求出应力分量，再求出应变和位移分量。
（2）采用应力求解时必须满足相容方程，是因为相容方程是应变 对应位移存在且连续的必要条件。当应变分量满足相容方程后，可以求 出对应的位移分量，说明位移是存在的而且必然连续。反之，不满足相 容方程的应变分量，不是物体中实际存在的，也求不出对应的位移。
(3)混合法：同时以某些位移分量和某些应力分量作为基本未知 量，用包含上述基本未知量的微分方程和边界条件求出这些基本未知 量，再用相关方程求出其它未知量。
（4）用位移表示的平衡微分方程： 用位移表示的应力边界条件：
位移边界条件：，（在上）
按应力求解平面问题，满足的平衡微分方程：
用应力表示的相容方程：应力边界条件
16简述按应力求解平面问题时的逆解法和半逆解法。 答：所谓逆解法，就是先按某种方法给出一组满足全部基本方程的应力 分量或位移分量，然后考察，在确定的坐标系下，对于形状和几何尺寸 完全确定的物体，当其表面受什么样的面力作用或具有什么样的位移 时，才能得到这组解答。所谓的半逆解法，就是针对所要求解的问题， 根据弹性体的几何形状、受力特点或材料力学已知的初等结果，假设一 部分应力分量或位移分量为已知，然后由基本方程求出其他量，把这些 量合在一起来凑合已知的边界条件；或者把全部的应力分量或位移分量 作为已知，然后校核这些假设的量是否满足弹性力学的基本方程和边界 条件。
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2007年同济大学博士入学考试
弹性力学
1、已知应力分量和截面的方向，求截面上的F στn 、、
2、已知应变分量，问其是否为弹性变形可能的情况
3、梁上受力如图所示，用瑞利－里兹法求梁上最大挠度max ω
q
4、告诉应力函数的形式，问是否可以作为薄板的解，并画出板上荷载的情况
x
5、已知开孔板上应力分量，(12cos2)q θσθ=-、0r σ=、0r θτ= 求圆孔边界上最大应力、最小应力
q
q
x
6、推导柱形杆扭转应力解法的公式，并解椭圆受扭
7、已知一点的应力情况为012111210⎡⎤
⎢⎥⎢⎥
⎢⎥⎣⎦
，求主应力、主应变、及相应的方向
8、用瑞利－里兹法解板，（已知挠度ω的表达式） 求挠度ω表达式中的系数11c
9、推导平面应力、平面应变问题的应变能密度并比较大小
10、简述艾里应力函数的涵义。