1、Unit 5 第五單元Structural Analysis 結(jié)構(gòu)分析1智囊經(jīng)驗(yàn)教學(xué)目標(biāo)了解結(jié)構(gòu)分析的假定和幾種方法了解結(jié)構(gòu)分析中運(yùn)用的幾個(gè)定理熟悉結(jié)構(gòu)力學(xué)中的專業(yè)詞匯熟悉科技類文獻(xiàn)中的常用句型熟悉as a consequence、 as a result、for these reasons、therefore 以及as a consequence of、as a result of 的含義；state 的不同用法；relate to、relate.to、be related to 的區(qū)別；be equal to、be equated to、equal 的用法； with respect to

2、、regarding、considering、as to 的用法2智囊經(jīng)驗(yàn) A structure consists of（由.組成）a series of connected parts used to support loads. Notable（顯著的） examples include buildings, bridges, towers, tanks, and dams. The process（過(guò)程）of creating any of these structures requires planning（規(guī)劃）, analysis, design, and constructio

3、n（施工）. Structural analysis consists of （包括）a variety of mathematical procedures（數(shù)學(xué)程序）for determining such quantities as the member forces and various structural displacements（位移） as a structure responds to its loads. Estimating realistic loads for the structure considering（根據(jù)）its use and location is

4、 often a part of structural analysis. 結(jié)構(gòu)由一系列相連的用以支撐荷載的構(gòu)件組成。顯著的例子包括建筑、橋梁、塔、水箱和大壩等。建造這些結(jié)構(gòu)中的任何一個(gè)的過(guò)程需要規(guī)劃、分析、設(shè)計(jì)和建造。結(jié)構(gòu)分析包括各種各樣的數(shù)學(xué)程序以確定諸如當(dāng)一個(gè)結(jié)構(gòu)對(duì)荷載有響應(yīng)時(shí)構(gòu)件的力和不同結(jié)構(gòu)位移的大小。根據(jù)結(jié)構(gòu)的使用和位置來(lái)估計(jì)它的實(shí)際荷載經(jīng)常是結(jié)構(gòu)分析的一部分。3智囊經(jīng)驗(yàn) Only two assumptions are made regarding（關(guān)于）the materials used in the structures of this chapter. First, th

5、e material has a linear stress-strain relationship（線性的應(yīng)力-應(yīng)變關(guān)系）. Second, there is no difference in the material behavior when stressed in tension vis-a-vis（與.相比）compression. The frames and trusses studied are plane structural systems（平面結(jié)構(gòu)體系）. It will be assumed that there is adequate bracing perpendi

6、cular to（垂直于）the plane so that no member will fail due to an elastic instability（彈性失穩(wěn)）. The very important consideration regarding such instability will be left for the specific（具體的）design course. 4智囊經(jīng)驗(yàn)All structures are assumed to undergo only small deformations as they are loaded. As a consequence

7、（因此）we assume no change in the position or direction of a force as a result of （由于）structural deflections（變位）. Finally, since linear elastic materials and small displacement are assumed, the principle of superposition will apply in all cases. Thus the displacements or internal forces that arise from

8、 two different forces systems applied one at a time（一次一個(gè)）may be added algebraically（幾何相加）to determine the structures response when both system(s) are applied simultaneously.5智囊經(jīng)驗(yàn) 關(guān)于本章結(jié)構(gòu)中所用的材料只作了兩點(diǎn)假設(shè)。首先，材料具有線性的應(yīng)力應(yīng)變關(guān)系。其次，材料的性能在受拉和受壓時(shí)沒(méi)有區(qū)別。研究的框架和桁架是平面結(jié)構(gòu)體系。假定垂直于平面的方向有足夠的支撐，因而構(gòu)件不會(huì)因?yàn)閺椥允Х€(wěn)而失效。一個(gè)非常重要的關(guān)于這種失穩(wěn)的考

9、慮留待具體的設(shè)計(jì)過(guò)程。假定所有的結(jié)構(gòu)在它們加荷時(shí)只經(jīng)歷小的變形。因此，我們假定當(dāng)結(jié)構(gòu)變位時(shí)荷載的位置與方向不變。最后，因?yàn)榧俣司€彈性材料和小位移，疊加原理將適用于所有的情況。這樣當(dāng)兩種不同的力系同時(shí)施加時(shí)，可以由不同的力系一次施加一個(gè)引起的位移或內(nèi)力幾何相加來(lái)確定結(jié)構(gòu)的響應(yīng)。6智囊經(jīng)驗(yàn) In the real sense（真正意義上）an exact analysis of a structure can never be carried out since estimates always have to be made of the loadings and the strength o

10、f the materials composing（構(gòu)成）the structure. Furthermore, points of application（作用點(diǎn)）for the loadings must also be estimated. It is important, therefore, that the structural engineers develop（形成）the ability to model（模擬）or idealize（使.理想化）a structure so that he or she can perform a practical force analy

11、sis of the members. 真正意義上對(duì)一個(gè)結(jié)構(gòu)準(zhǔn)確的分析是永遠(yuǎn)也不可能進(jìn)行的，因?yàn)榭偸遣坏貌还烙?jì)荷載和構(gòu)成結(jié)構(gòu)的材料的強(qiáng)度。而且，必須估計(jì)荷載的作用點(diǎn)。因此，結(jié)構(gòu)工程師有能力模擬一個(gè)結(jié)構(gòu)或使其理想化很重要，這樣，他或她能對(duì)構(gòu)件進(jìn)行實(shí)際的力的分析。7智囊經(jīng)驗(yàn) Structural members are joined together in various ways depending on the intent（意圖）of the designer. The two types of joints most often specified（規(guī)定的）are the pin conn

12、ection and the fixed joint（節(jié)點(diǎn)）. A pin-connected joint allows some freedom for slight（輕微）rotation, whereas the fixed joint allows no relative rotation between the connected members. In reality, however, all connections exhibit（顯現(xiàn)）some stiffness toward joint rotations, owing to friction（摩擦）and materia

13、l behavior. When selecting a particular model for each support（支座）or joint, the engineer must be aware of how the assumptions will affect the actual performance（運(yùn)行）of the member and whether the assumptions are reasonable for the structural design. In reality, all structural supports actually exert（產(chǎn)

14、生）distributed surface loads（面荷載）on their contacting members. 8智囊經(jīng)驗(yàn) The resultants（合力） of these load distributions are often idealized as the concentrated forces（集中力）and moments, since the surface area （表面積）over which the distributed load acts is considerably smaller than the total surface area of th

15、e connecting members. The ability to reduce an actual structure to（將.簡(jiǎn)化為）an idealized form can only be gained by experience. In engineering practice, if it becomes doubtful（不明確）as to how to model a structure or transfer the loads to the members, it is best to consider several idealized structures an

16、d loadings and then design the actual structure so that it can resist（抵抗）the loadings in all the idealized models.9智囊經(jīng)驗(yàn) 結(jié)構(gòu)構(gòu)件根據(jù)設(shè)計(jì)者的意圖采用不同的方式連在一起。最常規(guī)定的兩種節(jié)點(diǎn)是鉸接節(jié)點(diǎn)和固定節(jié)點(diǎn)。鉸接節(jié)點(diǎn)允許有一些輕微的轉(zhuǎn)動(dòng)自由，而固定節(jié)點(diǎn)不允許相連的構(gòu)件有相對(duì)的轉(zhuǎn)動(dòng)。但是，事實(shí)上由于摩擦和材料的特性使所有的連接對(duì)節(jié)點(diǎn)的轉(zhuǎn)動(dòng)顯現(xiàn)出一些剛度。當(dāng)為每一個(gè)支座或節(jié)點(diǎn)選擇一個(gè)特定的模型時(shí)，工程師必須知道該假設(shè)將如何影響構(gòu)件的實(shí)際運(yùn)行，以及該假設(shè)是否對(duì)結(jié)構(gòu)的設(shè)計(jì)是合理的。實(shí)

17、際上，所有的結(jié)構(gòu)支座在它們接觸的構(gòu)件上產(chǎn)生分布的面荷載。這些荷載分布的合力常常理想化為集中力和彎矩，因?yàn)榉植己奢d作用的表面面積比相連的構(gòu)件的總的表面面積小很多。將一個(gè)實(shí)際的結(jié)構(gòu)簡(jiǎn)化成一種理想的形式的能力只有通過(guò)經(jīng)驗(yàn)才能獲得。在工程實(shí)踐中，如果就怎樣模擬一個(gè)結(jié)構(gòu)或?qū)⒑奢d傳遞給構(gòu)件變得難以確定時(shí)，最好考慮幾個(gè)理想的結(jié)構(gòu)和荷載，然后設(shè)計(jì)實(shí)際的結(jié)構(gòu)，使它在所有理想的模型中都能抵抗荷載。10智囊經(jīng)驗(yàn) It may be recalled（回想）from statics that a structure or one of its members is in equilibrium（處于平衡） when

18、it maintains a balance of force and moment. When all the forces in a structure can be determined strictly from these equations, the structure is referred to as statically determinate（靜定的）. Structures having more unknown forces than available equilibrium equations（平衡方程）are called statically indetermi

19、nate. As a general rule, a structure can be identified as（確定）being either statically determinate or statically indeterminate by drawing free-body diagrams（隔離體圖）of all its members, or selective parts of its members, and then comparing the total number of unknown reactive force and moment components（分

20、量）with the total number of available equilibrium equations. 11智囊經(jīng)驗(yàn) 從靜力學(xué)可以回想起當(dāng)一個(gè)結(jié)構(gòu)或它的一個(gè)構(gòu)件維持力和彎矩的平衡時(shí)即處于平衡狀態(tài)。當(dāng)一個(gè)結(jié)構(gòu)中所有的力能嚴(yán)格地根據(jù)這些方程式來(lái)確定，該結(jié)構(gòu)稱為靜定的。如果結(jié)構(gòu)上未知的力比能得到的平衡方程多時(shí)稱為超靜定結(jié)構(gòu)。作為一般的規(guī)律，一個(gè)結(jié)構(gòu)可以通過(guò)畫出所有構(gòu)件或經(jīng)選擇的部分構(gòu)件的隔離體圖，然后比較未知的反力和彎矩的分量總數(shù)目與可用的平衡方程總數(shù)目是否相等來(lái)確定其是靜定結(jié)構(gòu)還是超靜定結(jié)構(gòu)。12智囊經(jīng)驗(yàn) In particular, if a structure is stati

21、cally indeterminate, the additional equations（附加方程）needed to solve for（求解）the unknown reactions（反力）are obtained by relating the applied loads and reactions to the displacement or slope（轉(zhuǎn)角）at different points on the structure. These equations, which are referred to as compatibility equations（相容性方程或協(xié)調(diào)

22、方程）, must be equal in number to the degree of indeterminacy（不確定次數(shù)）of the structure. Compatibility equations involve（涉及）the geometric and physical properties of the structure. 特別地，如果一個(gè)結(jié)構(gòu)是超靜定的，可以通過(guò)建立作用力和反力與結(jié)構(gòu)不同點(diǎn)上的位移或轉(zhuǎn)角的關(guān)系來(lái)得到用以求解未知反力所需的附加方程。這些稱為相容性方程的方程式在數(shù)量上必須等于結(jié)構(gòu)的不確定次數(shù)。相容性方程涉及結(jié)構(gòu)的幾何和物理性能。13智囊經(jīng)驗(yàn) There a

23、re two fundamental methods of analysis for trusses: the method of joints and the method of sections. Both start with（從.著手）a free-body diagram of the truss as a whole（基本上）, from which the equilibrium equations are written and solved for the support reactions（支座反力）. 有兩種分析桁架的基本方法：節(jié)點(diǎn)法和截面法。兩種方法基本上都從桁架的隔離

24、體圖著手，根據(jù)它可以寫出平衡方程并求解支座反力。14智囊經(jīng)驗(yàn) The method of joints: After the support reactions have been found, a joint is selected that has no more than（不超過(guò)）two members connecting for which the axial forces are unknown. The free-body diagram of that joint is drawn, the forces are summed（合計(jì)）in two directions, and

25、 each sum is equated to（等于）zero. When drawing the free-body diagram, it is a good idea to assume that the unknown forces are tensions and to show（表示）them so on the free-body diagram by their exerting a pull on（對(duì).施加拉力）the joint. When this is assumed, the resulting sign（符號(hào)）of the unknowns when evaluat

26、ed（計(jì)算）will match（符合）the conventional（習(xí)慣的）+ for tension and for compression. Once a joint has been analyzed, its members become knowns, and adjacent joints（相鄰節(jié)點(diǎn)）, which might have had three or more unknowns, can then be solved since some of these unknowns have become knowns. This process（過(guò)程）continues

27、 from joint to joint, each time selecting a joint whose number of unknown members does not exceed 2.15智囊經(jīng)驗(yàn)節(jié)點(diǎn)法：求出支座反力后，選擇一個(gè)節(jié)點(diǎn)其上連接著軸向力未知的構(gòu)件不超過(guò)兩根。畫出節(jié)點(diǎn)的隔離體圖，將力在兩個(gè)方向上進(jìn)行合計(jì)，每個(gè)方向（力）的合計(jì)等于零。當(dāng)畫出隔離體圖時(shí)，有個(gè)好主意是假定未知力是拉力，并在隔離體圖上通過(guò)對(duì)該節(jié)點(diǎn)施加一個(gè)拉力來(lái)表示。這樣假定后，未知力計(jì)算結(jié)果的符號(hào)將與習(xí)慣的正為拉力負(fù)為壓力相符。一旦一個(gè)節(jié)點(diǎn)已經(jīng)被分析，其上的構(gòu)件成為已知構(gòu)件，相鄰的節(jié)點(diǎn)可能曾經(jīng)有三個(gè)或更多的未

28、知力，但因?yàn)槠渲械囊恍┮呀?jīng)成為已知，因此也能求出。這個(gè)過(guò)程從一個(gè)節(jié)點(diǎn)到另一個(gè)節(jié)點(diǎn)連續(xù)進(jìn)行，每次選擇的節(jié)點(diǎn)其上未知構(gòu)件（力）的數(shù)量不超過(guò)兩根。16智囊經(jīng)驗(yàn) Almost all truss systems are configured（裝配）so that analysis using the method of joints must begin at one end and proceed（繼續(xù)）joint by joint toward the other end. If it is necessary to evaluate the forces carried by a member

29、located（位于）some distance from the ends, the method of joints requires the calculation of the forces in many members before the desired one is reached. The method of sections provides a means（方法）for a direct calculation in these cases. After the support reactions have been calculated the truss is cut

30、 through（切開(kāi)）(analytically分析上) so that one part of the truss is completely severed from the rest. When this is done, no more than three unknown members should be cut. If possible（如果可能）the cut（切口）should pass through the member or members whose internal forces are to be found. A free-body diagram of th

31、e part of the truss on one side of（在.一邊）this section is drawn, and the internal forces are found through the equilibrium equations. Since the system of forces（力系）on the free-body diagram is a plane non-concurrent（非共點(diǎn)）force system, three equilibrium equations may be written and solved for the three u

32、nknowns.17智囊經(jīng)驗(yàn) 幾乎所有的桁架體系是裝配的，因此采用節(jié)點(diǎn)法進(jìn)行的分析必須從一個(gè)端點(diǎn)開(kāi)始，并一個(gè)節(jié)點(diǎn)連著一個(gè)節(jié)點(diǎn)地朝另一個(gè)端點(diǎn)繼續(xù)進(jìn)行。如果有必要計(jì)算位于端部一定距離的構(gòu)件上的力，節(jié)點(diǎn)法需要在到達(dá)這根要求（計(jì)算）的構(gòu)件之前計(jì)算很多構(gòu)件中的力。在這些情況下截面法提供了一個(gè)直接計(jì)算的方法。當(dāng)求出支座反力后，桁架（在分析上）被切開(kāi)，從而一部分桁架同其余部分完全分離。當(dāng)這樣切開(kāi)時(shí)，應(yīng)該切出不超過(guò)三個(gè)構(gòu)件的力是未知的。如果可能，切口應(yīng)穿過(guò)將要求解內(nèi)力的構(gòu)件。畫出在截面一邊的桁架部分的隔離體圖，并通過(guò)平衡方程式求解內(nèi)力。由于隔離體圖上的力系是平面非共點(diǎn)的，因而可以寫出三個(gè)平衡方程式并求出三個(gè)未

33、知力。18智囊經(jīng)驗(yàn) Influence lines（影響線）have important application for（應(yīng)用）the design of structures that resist large live loads（活荷載）. An influence line represents（代表）the variation of either the reaction, shear, moment, or deflection at a specific （特定的）point in a member as concentrated force moves over the mem

34、ber. Once this line is constructed（作圖）, one can tell at a glance（一眼便知）where a live load should be placed on the structure so that it creates（引起）the greatest influence at the specified point. Furthermore, the magnitude（大?。﹐f the associated （相關(guān)的）reaction, shear, moment, or deflection at the point can

35、then be calculated from the ordinates（縱坐標(biāo)）of the influence-line diagram. 19智囊經(jīng)驗(yàn)For these reasons（因此）, influence lines play an important part in the design of bridges, industrial crane rails（吊車軌道）, conveyors, and other structures where loads move across their span（全長(zhǎng)）. Although the procedure（步驟）for c

36、onstructing an influence line is rather basic（基本的）, one should clearly be aware of the difference between constructing an influence line and constructing a shear or moment diagram. Influence lines represent the effect of a moving load only at a specified point on a member, whereas shear and moment d

37、iagrams represent the effect of fixed loads at all points along the axis of the member.20智囊經(jīng)驗(yàn) 影響線在設(shè)計(jì)抵抗大量活荷載的結(jié)構(gòu)時(shí)有著重要的應(yīng)用。一根影響線代表著當(dāng)集中力在構(gòu)件上移動(dòng)時(shí)構(gòu)件上一個(gè)特定點(diǎn)的反力、剪力、彎矩或撓度的變化。一旦畫出這根線，任何人一眼便知活荷載應(yīng)該置于結(jié)構(gòu)的哪個(gè)位置才能對(duì)這個(gè)特定的點(diǎn)引起最大的影響。而且，這點(diǎn)上相關(guān)的反力、剪力、彎矩或撓度可從影響線圖的縱坐標(biāo)上計(jì)算出來(lái)。因此，影響線在橋梁、工業(yè)吊車軌道、輸送機(jī)和其它有荷載在整個(gè)結(jié)構(gòu)長(zhǎng)度上移動(dòng)的結(jié)構(gòu)設(shè)計(jì)中扮演著重要的角色。雖然畫出一

38、條影響線的步驟是相當(dāng)基本的，但任何人應(yīng)該清楚地意識(shí)到畫一條影響線與畫一條剪力或彎矩圖的區(qū)別。影響線只代表著移動(dòng)荷載對(duì)構(gòu)件上特定點(diǎn)的影響，而剪力和彎矩圖代表固定荷載對(duì)沿著構(gòu)件軸線的所有點(diǎn)的影響。21智囊經(jīng)驗(yàn) Deflections of structures can occur from various sources（原因）, such as loads, temperature, fabrication errors, or settlement. In design, deflections must be limited in order to prevent cracking of a

39、ttached（附屬的） brittle materials such as concrete or plaster (石膏) . Furthermore, a structure must not vibrate or deflect（變位）severely in order to “appear” safe for its occupants（居住者）. More important, though（然而）, deflections at specified points in a structure must be computed if one is to analyze static

40、ally indeterminate structures. We often determine the elastic deflections of a structure using both geometrical and energy methods. Also, the methods of double integration（雙重積分）are used. The geometrical methods include the moment-area theorems（彎矩圖面積定理）and the conjugate-beam method（共軛梁法）, and the ene

41、rgy methods to be considered are based on virtual work（虛功）and Castiglianos theorem（卡氏最小功定理）. Each of these methods has particular advantages or disadvantages.22智囊經(jīng)驗(yàn) 結(jié)構(gòu)的撓度可以因不同的原因而發(fā)生，如荷載、溫度、制造錯(cuò)誤或沉降。設(shè)計(jì)中，撓度必須加以限制以阻止附屬的脆性材料如混凝土或石膏的開(kāi)裂。而且，為了向居住者顯示安全性，結(jié)構(gòu)不能嚴(yán)重地振動(dòng)或變位。而更重要的是如果有人要分析超靜定結(jié)構(gòu)，必須計(jì)算出結(jié)構(gòu)中規(guī)定點(diǎn)的撓度。我們通常采用幾何

42、法和能量法來(lái)確定結(jié)構(gòu)的彈性撓度。也采用雙重積分法。幾何法包括彎矩圖面積定理和共軛梁法，而考慮的能量法是基于虛功定理和卡式最小功定理。每一種方法都有其特別的優(yōu)缺點(diǎn)。23智囊經(jīng)驗(yàn) We can determine the equation of the elastic curve by integration of equation d2v / dx2 = M / EI. Solution of this equation requires two successive（連續(xù)的）integrations to obtain the deflection v of the elastic curve

43、. For each integration, it is necessary to introduce（引入）a “constant of integration”（積分常數(shù)）, and then solve for the constants to obtain a unique solution（唯一解）for a particular（特定的）problem. It should be realized that the method of double integration is suitable only for elastic deflections（變位）such that

44、the beams slope is very small. Furthermore, the method considers only deflections due to bending. 我們可以通過(guò)對(duì)方程d2v / dx2 = M / EI的積分來(lái)確定彈性曲線的方程。該方程的求解需要兩個(gè)連續(xù)的積分，以獲得彈性曲線的撓度v。對(duì)每次積分，有必要引入積分常數(shù)，求出該常數(shù)以獲得一個(gè)特定問(wèn)題的唯一解。應(yīng)該了解到雙重積分法只適合于彈性變位，因而梁的轉(zhuǎn)角是非常小的。而且，該法只考慮了由于彎曲引起的撓度。24智囊經(jīng)驗(yàn) The initial ideas（最初的概念）for the two momen

45、t-area theorems were developed（提出）by Otto Mohr and later stated formally（正式確定）by Charles E. Greene in 1872. These theorems provide a semi-graphical （半圖解）technique for determining the slope of the elastic curve and its deflection due to bending. They are particularly advantageous（有利）when used to solv

46、e problems involving beams especially those subjected to a series of concentrated loadings or having segments（段）with different moment of inertia（慣性矩）. Theorem 1: The change in slope（轉(zhuǎn)角變化）between any two points on the elastic curve equals the area of the M / EI diagram between these two points. Theor

47、em 2: The deviation（偏差）of the tangent（正切）at point B on the elastic curve with respect to（相對(duì)于）the tangent at point A equals the “moment” of the M / EI diagram between the two points A and B computed about point A (the point on the elastic curve), where（這里）the deviation tA/B is to be determined.25智囊經(jīng)驗(yàn)

48、 最初的關(guān)于兩個(gè)彎矩圖面積定理的概念是由Otto Mohr提出，后來(lái)由Charles E. Greene在1872年正式加以確定。這些定理為確定彈性曲線由于彎曲引起的轉(zhuǎn)角和撓度提供了半圖解的方法。當(dāng)用以解決包括梁在內(nèi)的問(wèn)題，特別是那些承受一組集中加載的梁或有著不同慣性矩的梁段時(shí)，它們（指彎矩圖面積定理）是特別得有利。定理1：彈性曲線上任何兩點(diǎn)之間轉(zhuǎn)角的變化等于這兩點(diǎn)之間的M / EI圖的面積。定理2：彈性曲線上B點(diǎn)的正切相對(duì)于A點(diǎn)的正切的偏差等于點(diǎn)A與點(diǎn)B之間的M / EI圖對(duì)A點(diǎn)（該點(diǎn)在彈性曲線上）的矩，這里偏差tA/B將被確定。26智囊經(jīng)驗(yàn) The conjugate-beam metho

49、d was first presented（提出）by Otto Mohr in 1860. Essentially（本質(zhì)上）, it requires the same amount of computation（計(jì)算量）as the moment-area theorems to determine a beams slope or deflection; however, this method relies only on the principles of statics and hence its application will be more familiar（常見(jiàn)）. The

50、 basis for the method comes from（來(lái)自于）the similarity（相似性）between both dV/dx = - and d2M/dx2 = - , which relate a beams internal shear and moment to its applied loading, and d/dx = M/EI and d2y/dx2 = M/EI, which relate the slope and deflection of its elastic curve to the internal moment. 共軛梁法首先在1860年由

51、Otto Mohr提出。本質(zhì)上說(shuō)，它與彎矩圖面積定理一樣在確定梁的轉(zhuǎn)角或撓度上需要相同的計(jì)算量；但是這種方法只依賴于靜力學(xué)的原理，因此，它的應(yīng)用更常見(jiàn)。該法的基礎(chǔ)來(lái)自于dV/dx=-和d2M/dx2=-之間的相似性，它將梁的內(nèi)部剪力和彎矩與它施加的荷載聯(lián)系起來(lái)，而d/dx = M/EI和d2y/dx2 = M/EI將彈性曲線的轉(zhuǎn)角和撓度與內(nèi)部彎矩聯(lián)系起來(lái)。27智囊經(jīng)驗(yàn) Note that the shear V compares with（與.對(duì)應(yīng)）the slope , the moment M compares with the displacement y and the intensi

52、ty of the external load compares with the area under the M/EI diagram. To make use of this comparison we will consider a beam having the same length as the real beam, but referred to here as the “conjugate beam”. 注意剪力V與轉(zhuǎn)角相對(duì)應(yīng)，彎矩M與位移y相對(duì)應(yīng)，而外力的強(qiáng)度與M/EI圖下的面積相對(duì)應(yīng)。為了利用這些對(duì)應(yīng)，我們將考慮一根與實(shí)際梁一樣長(zhǎng)的梁。但是這里稱為共軛梁。28智囊經(jīng)驗(yàn)In

53、 general, though（然而）, remember that if the real support allows a slope, the conjugate support must develop（產(chǎn)生）a shear; and if the real support allows a displacement, the conjugate support must develop a moment, note that the conjugate beam is “l(fā)oaded” with the M/EI diagram, in order to conform to（與.

54、一致）the load on the real beam. We can therefore state（陳述）two theorems related to the conjugate beam, namely（即）, Theorem 1: The slope at a point in the real beam is equal to the shear at the corresponding point（相應(yīng)點(diǎn)）in conjugate beam. Theorem 2: The displacement of a point in the real beam is equal to

55、the moment at the corresponding point in the conjugate beam.29智囊經(jīng)驗(yàn) 然而通常要記住如果實(shí)際的支座允許一個(gè)轉(zhuǎn)角，共軛的支座必須產(chǎn)生一個(gè)剪力；如果實(shí)際的支座允許一個(gè)位移，共軛的支座必須產(chǎn)生一個(gè)彎矩，注意共軛梁用M/EI圖來(lái)加荷，以便與實(shí)際梁上的荷載一致。因此，我們可以陳述與共軛梁相關(guān)的兩種定理，即，定理1：實(shí)際梁上某一點(diǎn)的轉(zhuǎn)角等于共軛梁上相應(yīng)點(diǎn)的剪力。定理2：實(shí)際梁上某一點(diǎn)的位移等于共軛梁上相應(yīng)點(diǎn)的彎矩。30智囊經(jīng)驗(yàn) For more complicated loadings or for structures such as tr

56、usses and frame, it is suggested（建議）that energy methods be used for the computation. All energy methods are based on the conservation of energy principle（能量守恒原則）, which states（規(guī)定）that the work（功）done by all the external forces acting on a structure, Ue, is transformed into（轉(zhuǎn)化為）internal work or strai

57、n energy（應(yīng)變能）UI, which is developed（形成）when the structure deforms（變形）. 對(duì)于較復(fù)雜的荷載或結(jié)構(gòu)如桁架和框架，建議應(yīng)該采用能量法來(lái)計(jì)算。所有的能量法是基于能量守恒原則，它規(guī)定了作用在結(jié)構(gòu)上的所有外力作的功Ue轉(zhuǎn)化成內(nèi)部功或結(jié)構(gòu)變形時(shí)形成的應(yīng)變能UI 。31智囊經(jīng)驗(yàn) The principle of virtual work was developed by John Bernoulli in 1717 and is sometimes referred to as the unit-load method（單位荷載法）. It

58、 provides a general means（一般方法）of obtaining the displacement and slope at a point on a structure, be it（無(wú)論是）a beam, frame, or truss. Before developing the principle of virtual work, it is necessary to make some general statements（一般規(guī)定）regarding the principle of work and energy. 虛功原理在1717年由John Berno

59、ulli提出，有時(shí)稱為單位荷載法。它提供了獲得結(jié)構(gòu)上某一點(diǎn)的位移和轉(zhuǎn)角的一般的方法，不管該結(jié)構(gòu)是梁、框架還是桁架。在提出虛功原理之前，關(guān)于功和能量的原理有必要作些一般規(guī)定。32智囊經(jīng)驗(yàn)If we take（?。゛ deformable（可變形）structure of any shape or size and apply a series of external loads P to it, it will cause internal loads u at points throughout the structure. It is necessary that the external

60、and internal loads be related by the equation of equilibrium. As a consequence of（通過(guò)）these loadings, external displacement will occur at the P loads and internal displacement will occur at each point of internal load u. 如果我們?nèi)∫粋€(gè)任何形狀或尺寸的可變形結(jié)構(gòu)，并對(duì)它施加一組外力P，它將導(dǎo)致整個(gè)結(jié)構(gòu)上的點(diǎn)產(chǎn)生內(nèi)力u。有必要通過(guò)平衡方程將內(nèi)外力聯(lián)系起來(lái)。通過(guò)這些荷載，外部的位移發(fā)