英漢雙語彈性力學(xué)2

1、1,Chapter 2 The Basic Theory of the Plane Problem,2,第二章 平面問題的基本理論,3,The Basic Theory of the Plane Problem,Chapter 2 The Basic theory of the Plane Problem,2-11 Stress function.Inverse solution method and semi-inverse method,2-1 Plane stress problem and plane strain problem,2-2 Differential equation o

2、f equilibrium,2-3 The stress on the incline.Principal stress,2-4 Geometrical equation.The displacement of the rigid body,2-5 Physical equation,2-6 Boundary conditions,2-7 Saint-Venants principle,2-8 Solving the plane problem according to the displacement,2-9 Solving the plane problem according to th

3、e stress.Compatible equation,2-10 The simplification under the circumstances of ordinary physical force,Exercise Lesson,4,平面問題的基本理論,第二章 平面問題的基本理論,2-11 應(yīng)力函數(shù)逆解法與半逆解法,2-1 平面應(yīng)力問題與平面應(yīng)變問題,2-2 平衡微分方程,2-3 斜面上的應(yīng)力主應(yīng)力,2-4 幾何方程剛體位移,2-5 物理方程,2-6 邊界條件,2-7 圣維南原理,2-8 按位移求解平面問題,2-9 按應(yīng)力求解平面問題。相容方程,2-10 常體力情況下的簡化,習(xí)題課,

4、5,1.Plane stress problem,2-1 Plane stress problem and plane strain problem,In actual problem,it is strictly saying that any elastic body whose external force for suffering is a space system of forces is generally the space object.However,when both the shape and force circumstance of the elastic body

5、 for investigating have their own certain characteristics.As long as the abstraction of the mechanics is handled together with appropriate simplification,it can be concluded as the elasticity plane problem. The plane problem is divided into the plane stress problem and plane strain problem.,Equal th

6、ickness lamella bears the surface force that parallels with plate face and dont change along the thickness.At the same time,so does the volumetric force. z = 0 zx = 0 zy = 0,Fig.21,The Basic Theory of the Plane Problem,6,一、平面應(yīng)力問題,2-1 平面應(yīng)力問題與平面應(yīng)變問題,在實(shí)際問題中，任何一個(gè)彈性體嚴(yán)格地說都是空間物體，它所受的外力一般都是空間力系。但是，當(dāng)所考察的彈性體的

7、形狀和受力情況具有一定特點(diǎn)時(shí)，只要經(jīng)過適當(dāng)?shù)暮喕土W(xué)的抽象處理，就可以歸結(jié)為彈性力學(xué)平面問題。 平面問題分為平面應(yīng)力問題和平面應(yīng)變問題。,等厚度薄板，板邊承受平行于板面并且不沿厚度變化的面力，同時(shí)體力也平行于板面并且不沿厚度變化。 z = 0 zx = 0 zy = 0,圖21,平面問題的基本理論,7,The Basic Theory of the Plane Problem,Characteristics：,1) The dimension of length and breadth is far larger than that of thickness.,2) The force al

8、ong the plate face for suffering is the face force in parallel with plate face,and along the thickness even,the volumetric force is in parallel with plate force and doesnt change along the thickness, and has no external force function on the surface front and back of the flat panel.,Attention: Plane

9、 stress problem z =0,but ,this is contrary to plane strain problem.,8,平面問題的基本理論,特點(diǎn)：,1) 長、寬尺寸遠(yuǎn)大于厚度,2) 沿板邊受有平行板面的面力，且沿厚度均布，體力,平行于板面且不沿厚度變化，在平板的前后表面上,無外力作用。,9,2.Plane strain problem,Very long column bears the face force in parallel with plate face and doesnt change along the length on the column face,a

10、t the same time,so does the volumetric force. z = 0 zx = 0 zy = 0,x,Fig. 22,The Basic Theory of the Plane Problem,For example:dam,circular cylinder piping by the internal air pressure and long level laneway etc.,Attention: Plane strain problemz = 0，but ,this is contrary to plane stress problem.,10,二

11、、平面應(yīng)變問題,很長的柱體，在柱面上承受平行于橫截面并且不沿長度變化的面力，同時(shí)體力也平行于橫截面并且不沿長度變化。 z = 0 zx = 0 zy = 0,x,圖 22,平面問題的基本理論,如：水壩、受內(nèi)壓的圓柱管道和長水平巷道等。,11,2-2 Differential Equation of Equilibrium,Whether plane stress problem or plane strain problem, is the research problem in plane xy,all the physics quantity has nothing to do with

12、z.,Discuss below the correlation between any point stress and volumetric force when the object is placed in the state of equilibrium,and lead an equilibrium differential equation from here.From the lamella shown in Fig.2-1,we take out a small and positive parallelepiped PABC,and take for an unit len

13、gth in the directional dimension in z.,Establishing the function of the positive stress force in an unit on the left side is ,the coordinate on the right side x gets the increment ,the positive stress on the face is ,spreading the formula above will be Taylors series:,The Basic Theory of the Plane P

14、roblem,12,2-2 平衡微分方程,平面問題的基本理論,13,After omitting small quantity of the two rank and above the two rank,can get ,at the same time, , , are get the state of stress from the drawing show.,While considering the volumetric force to the plane stress state,still prove mutual and equal theory of shearing st

15、rength.Regard the center D and straight line in parallel with the shaft of z as the moment shaft, list the equilibrium equation of the moment shaft :,The Basic Theory of the Plane Problem,14,略去二階及二階以上的微量后便得 同樣 、 、 都一樣處理，得到圖示應(yīng)力狀態(tài)。,對平面應(yīng)力狀態(tài)考慮體力時(shí)，仍可證明剪應(yīng)力互等定理。以通過中心D并平行于z軸的直線為矩軸，列出力矩的平衡方程 ：,將上式的兩邊除以 得到：,平

16、面問題的基本理論,15,Deduce the equilibrium differential equation of the plane stress problem below,list the equilibrium equation to the unit:,The Basic Theory of the Plane Problem,16,下面推導(dǎo)平面應(yīng)力問題的平衡微分方程，對單元體列平衡方程：,平面問題的基本理論,17,Sorting them gets:,The Basic Theory of the Plane Problem,18,整理得:,平面問題的基本理論,19,2-3 T

17、he stress on the Inclined Plane.Principal stress,1.The stress on the inclined plane Having known the stress weight of any point P inside the elastic body,we try to get the stress which pass the point P on the arbitrarily inclined cross section.From neighborhood of point P taking a plane AB,which is

18、in parallel with the inclined plane above,and draws a small set square or three column PAB on two planes which pass point P and have perpendicularity in the shaft of x and y.When the plane AB approaches point P infinitely,the mean stress on the plane AB will become the stress on the inclined plane a

19、bove.,Establish the length of the face AB in the plane xy is dS,N is the exterior normal direction,and its direction cosine is:,The Basic Theory of the Plane Problem,20,2-3 斜面上的應(yīng)力、主應(yīng)力,一、斜面上的應(yīng)力 已知彈性體內(nèi)任一點(diǎn)P處的應(yīng)力分量 ，求經(jīng)過該點(diǎn)任意斜截面上的應(yīng)力。為此在P點(diǎn)附近取一個(gè)平面AB，它平行于上述斜面，并與經(jīng)過P點(diǎn)而垂直于x軸和y軸的兩個(gè)平面劃出一個(gè)微小的三角板或三棱柱PAB。當(dāng)平面AB與P點(diǎn)無限接近

20、時(shí)，平面AB上的應(yīng)力就成為上述斜面上的應(yīng)力。,設(shè)AB面在xy平面內(nèi)的長度為dS，厚度為一個(gè)單位長度，N 為該面的外法線方向，其方向余弦為：,平面問題的基本理論,21,The projection of the whole stress on the inclined plane AB is XN and YN respectively along with the shaft of x and y.From the PAB equilibrium term can get:,Divide and get:,Same from and get:,The positive stress on th

21、e inclined plane AB,from the projection can get:,The shearing strength on the inclined plane AB,from the projection can get:,The Basic Theory of the Plane Problem,22,斜面AB上全應(yīng)力沿x軸及y軸的投影分別為XN和YN。由PAB的平衡條件 可得：,除以 即得：,同樣由 得出：,斜面AB上的正應(yīng)力 ，由投影可得：,斜面AB上的剪應(yīng)力 ，由投影可得：,平面問題的基本理論,23,3.Principal stress,If the shea

22、ring stress of some inclined plane through point P is equal to zero,then the positive stress of that inclined plane calls a principal stress of point P,but that inclined plane calls the main plane of the stress at point P,and the normal direction of that inclined plane calls the main direction of th

23、e stress at point P.,1.The size of the principal stress,2.The direction of the principal stress is in the perpendicularity with for each other.,The Basic Theory of the Plane Problem,24,二、主應(yīng)力,如果經(jīng)過P點(diǎn)的某一斜面上的切應(yīng)力等于零，則該斜面上的正應(yīng)力稱為P點(diǎn)的一個(gè)主應(yīng)力，而該斜面稱為P點(diǎn)的一個(gè)應(yīng)力主面，該斜面的法線方向稱為P點(diǎn)的一個(gè)應(yīng)力主向。,1.主應(yīng)力的大小,2.主應(yīng)力的方向 與 互相垂直。,平面問題的基

24、本理論,25,2-4 Geometrical Equation. The Displacement of the Rigid Body,In plane problem,every point inside the elastic body can produce the arbitrarily directional displacement.Take an unit PAB through any point P inside the elastic body,such as Fig.2-5 show.After the elastic body suffers force,the poi

25、nt P,A,B move to the point P、A、Brespectively.,Fig.25,一、The positive strain at point P,Here because of small deformation, PA for causing stretch and shrink from the y direction displacement v is the small quantity of a high rank and this small quantity may be omitted.,The Basic Theory of the Plane Pr

26、oblem,26,2-4 幾何方程、剛體位移,在平面問題中，彈性體中各點(diǎn)都可能產(chǎn)生任意方向的位移。通過彈性體內(nèi)的任一點(diǎn)P，取一單元體PAB，如圖2-5所示。彈性體受力以后P、A、B三點(diǎn)分別移動(dòng)到P、A、B。,圖25,一、P點(diǎn)的正應(yīng)變,在這里由于小變形，由y方向位移v所引起的PA的伸縮是高一階的微量，略去不計(jì)。,平面問題的基本理論,27,The same can get:,2.Shearing strain at point P,The corner of the line segment PA:,The same can get the corner of the line segment P

27、B:,Thus,The Basic Theory of the Plane Problem,28,同理可求得：,二、P點(diǎn)的切應(yīng)變,線段PA的轉(zhuǎn)角：,同理可得線段PB的轉(zhuǎn)角：,所以,平面問題的基本理論,29,Therefore get the geometrical equation of the plane problem,From the geometrical equation above,when the displacement weight of the object is completely certain,the deformation weight is completely

28、 certain,unique weight can not be made sure thoroughly.,The Basic Theory of the Plane Problem,30,因此得到平面問題的幾何方程：,由幾何方程可見，當(dāng)物體的位移分量完全確定時(shí)，形變分量即可完全確定。反之，當(dāng)形變分量完全確定時(shí)，位移分量卻不能完全確定。,平面問題的基本理論,31,2-5 The Physical Equation,In the isotropy of the complete elasticity,the relation between the deformation weight an

29、d the stress weight is established according to the Hookes law as follows:,The Basic Theory of the Plane Problem,32,2-5 物理方程,在完全彈性的各向同性體內(nèi)，形變分量與應(yīng)力分量之間的關(guān)系根據(jù)虎克定律建立如下：,平面問題的基本理論,33,Inside the formula,the E is a modulus of elasticity;the G is a stiffness modulus;the u is a poisson ratio.The relation of t

30、hree ones above:,1.The physics equation of the plane stress problem,And have:,the Basic Theory of the Plane Problem,34,式中，E為彈性模量；G為剛度模量； 為泊松比。三者的關(guān)系：,一、平面應(yīng)力問題的物理方程,且有：,平面問題的基本理論,35,2.The physics equation of the plane strain problem,3.The transformation relation of the relation type between the stress

31、 strain and the plane strain.,The relation type of the plane stress:,The Basic Theory of the Plane Problem,36,二、平面應(yīng)變問題的物理方程,三、平面應(yīng)力的應(yīng)力應(yīng)變關(guān)系式與平面應(yīng)變的關(guān)系式之間的 變換關(guān)系,將平面應(yīng)力中的關(guān)系式：,平面問題的基本理論,37,For change,Can get the relation type in the plane strain:,Because of the similarity of this kind,while solving plane st

32、rain problem,the corresponding equation of the plane problem and the elastic constant in the answer can be exchanged as above,can get the solution of the homologous plane strain problem.,The Basic Theory of the Plane Problem,38,作代換,就可得到平面應(yīng)變中的關(guān)系式：,由于這種相似性，在解平面應(yīng)變問題時(shí)，可把對應(yīng)的平面應(yīng)力問題的方程和解答中的彈性常數(shù)進(jìn)行上述代換，就可得到相

33、應(yīng)的平面應(yīng)變問題的解。,平面問題的基本理論,39,2-6 Boundary Conditions,When the object is placed in the state of equilibrium,its internal state of stress at all point should satisfy the equilibrium differential equation and also satisfy the boundary term on the boundary. According to the difference of the boundary condit

34、ion,the elasticity problem is divided into the displacement boundary problem,stress boundary problem and mixed boundary problem.,1.Displacement Boundary Term,When the displacement has been known on the boundary,the displacement of the point on the object boundary and the equal term of the fixed disp

35、lacement should be established.For example,if making the boundary of the fixed displacement is ,and have(on the ):,Among them, and means the displacement weight on the boundary,however, and is the coordinate function we have know the boundary.,The Basic Theory of the Plane Problem,40,2-6 邊界條件,當(dāng)物體處于平

36、衡狀態(tài)時(shí),其內(nèi)部各點(diǎn)的應(yīng)力狀態(tài)應(yīng)滿足平衡微分方程；在邊界上應(yīng)滿足邊界條件。 按照邊界條件的不同，彈性力學(xué)問題分為位移邊界問題、應(yīng)力邊界問題和混合邊界問題。,一、位移邊界條件,平面問題的基本理論,41,2.Stress boundary term,When the boundary of the object is given to surface force,then the stress of the object on the boundary should satisfy the equilibrium term of forces with the equilibrium of the

37、 surface force.,Among them, and are the surface force weights and , , , are the stress weights on the boundary.,When the boundary face is in perpendicularity in shaft x,stress boundary term can be changed briefly into:,When the boundary face is in perpendicularity in shaft y,stress boundary term can

38、 be changed briefly into:,The Basic Theory of the Plane Problem,42,二、應(yīng)力邊界條件,當(dāng)物體的邊界上給定面力時(shí)，則物體邊界上的應(yīng)力應(yīng)滿足與面力相平衡的力的平衡條件。,其中 和 為面力分量， 、 、 、 為邊界上的應(yīng)力分量。,當(dāng)邊界面垂直于 軸時(shí)，應(yīng)力邊界條件簡化為：,當(dāng)邊界面垂直于 軸時(shí)，應(yīng)力邊界條件簡化為：,平面問題的基本理論,43,3.Mixed boundary condition,1.The displacement has been known on a part of boundaries of the object

39、,the result of which have the displacement boundary term,the boundaries of other parts have the surface force we have know.And then there should be stress boundary term and displacement boundary term respectively on two parts of the boundaries.The left surface of the cantilever contains displacement

40、 boundary term,such as shown in Fig.2-6.,Top and bottom surface contains stress boundary term:,The right surface contains stress boundary term:,Fig.2-6,The Basic Theory of the Plane Problem,44,三、混合邊界條件,1.物體的一部分邊界上具有已知位移，因而具有位移邊界條件，另一部分邊界上則具有已知面力。則兩部分邊界上分別有應(yīng)力邊界條件和位移邊界條件。如圖2-6，懸臂梁左端面有位移邊界條件：,上下面有應(yīng)力邊界條

41、件：,右端面有應(yīng)力邊界條件：,圖2-6,平面問題的基本理論,45,2.On the same boundary,there are not only stress boundary term but displacement boundary term.Coupler sustains the boundary term,such as shown in Fig.2-7.,The alveolus boundary term shown in Fig.2-8.,Fig.2-7,The Basic Theory of the Plane Problem,46,2.在同一邊界上，既有應(yīng)力邊界條件又

42、有位移邊界條件。 如圖2-7連桿支撐邊界條件：,如圖2-8齒槽邊界條件：,圖2-7,平面問題的基本理論,47,2-7 Saint-Venant Principle,1.Saint-Venants Principle,If transforming a small part of the surface force on the boundary into the surface force that has equal effect but different distribution(The main vector is equal,so is the main quadrature to

43、the same point as well),and then the distribution of the stress force nearby will have prominent changes,but the influence from the distant place can not be accounted.,2.Give Examples,Establishing the component of the column forms,the centroid of area in cross sections of both ends suffers the tensi

44、ble force which is equal in size but contrary in direction,such as shown in Fig.2-9a.If transforming an or both ends of tensile force into the force at the same effect as the static force,such as shown in Fig.2-9b or Fig.2-9c,the distribution of stress force drawn only by broken line has prominent c

45、hanges,whereas,the influence of the rest parts can not be accounted.If changing both ends of tensile force into that of uniform distribution again,the gathering degree is equal to P/A and among them A is the cross-section area of the component,such as shown in Fig.2-9d,there is still the stress clos

46、e to both ends under the noticeable influence.,The Basic Theory of the Plane Problem,48,2-7 圣維南原理,一、圣維南原理,如果把物體的一小部分邊界上的面力，變換為分布不同但靜力等效的面力（主矢量相同，對于同一點(diǎn)的主矩也相同），那么，近處的應(yīng)力分布將有顯著的改變，但是遠(yuǎn)處所受的影響可以不計(jì)。,二、舉例,設(shè)有柱形構(gòu)件，在兩端截面的形心受到大小相等而方向相反的拉力 ，如圖2-9a。如果把一端或兩端的拉力變換為靜力等效的力，如圖2-9b或2-9c，只有虛線劃出的部分的應(yīng)力分布有顯著的改變，而其余部分所受的影響是可

47、以不計(jì)的。如果再將兩端的拉力變換為均勻分布的拉力，集度等于 ，其中 為構(gòu)件的橫截面面積，如圖2-9d，仍然只有靠近兩端部分的應(yīng)力受到顯著的影響。,平面問題的基本理論,49,Under the four kinds of circumstances above,parts of distribution of stress force distant from both ends have no marked difference.,Attention:,The application of the Saint-Venants principle is by no means separated

48、 from the term of Equal Effect of Static Force.,The Basic Theory of the Plane Problem,50,在上述四種情況下，離開兩端較遠(yuǎn)的部分的應(yīng)力分布，并沒有顯著的差別。,注意：,應(yīng)用圣維南原理，絕不能離開“靜力等效”的條件。,平面問題的基本理論,51,2-8 Solving the Plane Problem according to the displacement,There are three kinds of basic methods to solve the problem in elasticity:th

49、e solution to the problem according to displacement,stress force and admixture.,While solving problems using displacement method,we regard displacement weight as the basic function unknown.After getting displacement weight from only including the differential equation and boundary term of the displa

50、cement weight,then get the deformation weight using geometrical equation, therefore, get the stress weight with the physics equation.,1.Plane Stress Problem,In plane stress problem, the physics equation is:,The Basic Theory of the Plane Problem,52,2-8 按位移求解平面問題,在彈性力學(xué)里求解問題，有三種基本方法：按位移求解、按應(yīng)力求解和混合求解。,按

51、位移求解時(shí)，以位移分量為基本未知函數(shù)，由一些只包含位移分量的微分方程和邊界條件求出位移分量以后，再用幾何方程求出形變分量，從而用物理方程求出應(yīng)力分量。,一、平面應(yīng)力問題,在平面應(yīng)力問題中，物理方程為：,平面問題的基本理論,53,From three formulas above mentioned to solve the stress weight,can get: with the substitution of geometrical equation,we can get the elasticity equation:,Again equilibrium differential e

52、quation with substitution in formula(a), simplification hereafter, can get:,（a),This is the equilibrium differential equation to mean with the displacement, ie, when solving the plane stress problem according to displacement method, we adopt a basic differential equation for needs.,（1）,The Basic The

53、ory of the Plane Problem,54,由上列三式求解應(yīng)力分量，得：,將幾何方程代入，得彈性方程：,再將式（a）代入平衡微分方程，簡化以后，即得：,（a),這是用位移表示的平衡微分方程，也就是按位移求解平面應(yīng)力問題時(shí)所需用的基本微分方程。,（1）,平面問題的基本理論,55,The stress boundary term with substitution in formula(a), simplification hereafter, can get:,This is the stress force boundary to mean with the displacemen

54、t, ie, we adopt the boundary term of the stress force when solving the plane stress problem according to displacement method.,（2）,Sum up, when solving the plane stress problem according to displacement method, we should make the displacement weight satisfy differential equation(1) and combine to sat

55、isfy displacement boundary term or stress boundary term or stress boundary term(2) on the boundary. After getting displacement weight, we can get the deformation weight with geometrical equation and then get the stress force weight with the physics equation.,2.Plane strain problem,Make the substitut

56、ion between and in each equation of the plane strain problem:,The Basic Theory of the Plane Problem,56,將（a）式代入應(yīng)力邊界條件，簡化以后，得：,這是用位移表示的應(yīng)力邊界條件，也就是按位移求解平面應(yīng)力問題時(shí)所用的應(yīng)力邊界條件。,（2）,總結(jié)起來，按位移求解平面應(yīng)力問題時(shí)，要使得位移分量滿足微分方程（1），并在邊界上滿足位移邊界條件或應(yīng)力邊界條件（2）。求出位移分量以后，用幾何方程求出形變分量，再用物理方程求出應(yīng)力分量。,二、平面應(yīng)變問題,只須將平面應(yīng)力問題的各個(gè)方程中 和 作代換：,平面問題

57、的基本理論,57,2-9 Solving the Plane Problem According to the Stress Force.Compatible Equantion,While solving the plane problem according to the displacement, we must combine two partial differential equation of the second ranks to solve the problem, this is very difficult on the mathematics. But while so

58、lving the plane problem according to the stress force, we can avoid this difficulty and so what we adopt more is to get the solution according to the stress force.,While getting the solution according to the stress force, we regard stress weight as the basic function unknown.After getting displacement weight from only including the differential equation and boundary term of displacement weight, then get the deformation weight using physics equation, therefore, get the displacement weight with geometrical equation.,Compatible Equation,From geometrical equation of the plane problem:,The B