2平面問題基本理論

1、 2 Theory of Plane Problems21 Plane Stress and Plane Strain22 Differential Equations of Equilibrium23 Geometrical Equations. Rigid-body Displacement24 Physical Equations25 Stress at a Point26 Boundary Conditions. Saint-Venants Principle27 Solution of Plane Problem in Terms of Displacements28 Solutio

2、n of Plane Problem in Terms of Stresses29 Case of Constant Body Forces. Stress Function*SOLUTION OF PLANE PROBLEM*For the solution of an elasticity problem, we can proceed in three different ways:1. Take the displacement components as the basic unknown functions, formulate a system of differential e

3、quations and boundary conditions containing the displacement components only, solve for these unknown functions and thereby find the strain components by the geometrical equations and then the stress components by the physical equations. 2. Take the stress components as the basic unknown functions,

4、formulate a system of differential equations and boundary conditions containing the stress components only, solve for these unknown functions and thereby find the strain components by the physical equations and then the displacement components by the geometric equation.3. Take some of the displaceme

5、nt components and also some of the stress components as the basic unknown functions, formulate a system of differential equations and boundary conditions containing the stress components only, solve for these unknown functions and thereby find the other unknown functions.27 按位移求解平面問題平面問題的基本未知量有x，y，x

6、y，x，y，xy，u，v，根據(jù)基本方程即可求解。Solution of Plane Problem in Terms of Displacements求解方法有：按位移求解；按應(yīng)力求解；混合求解 P26按位移求解：以位移分量為基本函數(shù)，由只含位移分量的微分方程和邊界條件求出位移分量后，再求其他的未知量。Take the displacement components as the basic unknown function，formulate a system of differential equations and boundary conditions containing the d

7、isplacement components only，solve for these unknown functions and thereby find the strain components by the geometrical equations and then the stress components by the physical equations. 導(dǎo)出按位移求解的微分方程和邊界條件 1、微分方程 (differential equations) Formulate the differential equations and boundary conditions f

8、or the solution of a plane problem in terms of displacements 幾何方程：geometrical equations物理方程（平面應(yīng)力問題）physical equations（plane stress problem）將幾何方程代入物理方程Substitution of geometric Eqs. into these equations將上述方程代入平衡微分方程N(yùn)ow, using these relations in equilibrium equations按位移求解的平衡微分方程 拉密方程在平面問題中的應(yīng)用The diffe

9、rential equations for the solution of the problem in terms of displacements. 2、邊界條件(Boundary conditions) lx+mxy=Xmy+lxy=Y平面問題的應(yīng)力邊界條件（Stress boundary conditions of a plane problem）按位移求解時(shí)的應(yīng)力邊界條件為：用位移表示的應(yīng)力邊界條件。We obtain the stress boundary conditions of the problem in terms of displacements.歸納(To sum u

10、p), 按位移求解平面問題，要使位移分量滿足拉密方程和邊界條件，求出位移后，可用物理方程求應(yīng)力，用幾何方程求變形。The displace components u(x,y), v(x,y) in a plane stress problem must satisfythroughout the body considered and also satisfyon the surface of the body.For a plane strain problem, it is necessary in above equations. 對(duì)平面應(yīng)變問題，只許將上述方程中的 ；將 即可。28 按

11、應(yīng)力求解平面問題 相容方程 P28Solution of Plane Problem in Terms of Stresses按應(yīng)力求解：以應(yīng)力分量為基本函數(shù)，由只含應(yīng)力分量的平衡微分方程和相容方程及邊界條件求出應(yīng)力分量后，再求其他的未知量。Take the stress components as the basic unknown function，formulate a system of differential equations and boundary conditions containing the stress components only，solve for these

12、 unknown functions and thereby find the strain components by the physical equations and then the displacement components by the geometrical equations.含應(yīng)力分量，需保留The two differential equations of equilibrium contain the stress components only and may be used for their solution. 需建立補(bǔ)充方程 相容方程The third di

13、fferential equation can be obtained from the geometrical and physical equations . 相容方程（變形協(xié)調(diào)方程） Compatibility equation 1、平面問題的幾何方程The geometrical equations of a plane problem are2、將x對(duì)y的二階導(dǎo)數(shù)和y對(duì)x的二階導(dǎo)數(shù)相加 Adding the second derivative of x with respect to y and the second derivative of y with respect to x

14、, we get 相容方程Compatibility equation應(yīng)變分量x、y和xy必須滿足這個(gè)方程，才能保證位移分量u，v的存在。 若所選的x、y和xy不滿足這個(gè)方程，那么，由幾何方程中的任意兩個(gè)所求出的位移分量，將不滿足第三個(gè)方程。例如選x=0，y=0，xy=cxy 不滿足相容方程由此應(yīng)變求位移The compatibility equation for strain must be satisfied by the strain components x, y and xy to ensure the existence of single-valued continuous fu

15、nctions u and v connected with the strain components by the geometrical equations.第三個(gè)方程不能滿足，所求u，v不存在 用應(yīng)力表示的相容方程 Compatibility equation in terms of strainBy using physical equations, the compatibility equation can be transformed into a relation between the stress components. 將物理方程代入 平面應(yīng)力問題 for a plan

16、e stress problemSubstitution of the physical equations into簡(jiǎn)化上述相容方程（利用平衡方程） To transform this equation into a different form more suitable for use, we eliminate the term involving xy by using the differential equations of equilibrium. 用應(yīng)力表示的相容方程the compatibility equation in terms of stresses.Differe

17、ntiating the first equation with respect to x and the second with respect to y, adding them up and noting that xy=yx, we get(將兩式分別對(duì)x及y求導(dǎo)，并相加得)(將其代入相容方程，并簡(jiǎn)化后，得)Substituting this into the compatibility equation and performing some simplification, we obtain平面應(yīng)變問題的相容方程Compatibility equation in terms of

18、strainCompatibility equation for a plane strain problemFor a plane strain problem, an equation similar to above equation may be obtained simply byThe result is 歸納：1、按應(yīng)力求解平面問題，要求應(yīng)力分量必須滿足平衡微分方程和相應(yīng)的相容方程，在邊界上還要滿足應(yīng)力邊界條件（P29）。In the solution of a plane problem in terms of stresses，the stress components mu

19、st satisfy the differential equations of equilibrium and compatibility equation in the case of place strain. Besides,they must satisfy the stress boundary conditions.2、由于位移邊界條件無法用應(yīng)力分量或其導(dǎo)數(shù)來表示，所以對(duì)位移邊界條件或混合邊界條件，不可能按應(yīng)力求解得出精確解。Since the displacement boundary conditions can be expressed neither in terms o

20、f stress components nor in terms of their derivatives with respect to the coordinates,displacement boundary problems and mixed boundary problems cannot be solved in terms of stresses.對(duì)應(yīng)力邊界問題，應(yīng)力分量滿足了平衡微分方程、相應(yīng)的相容方程和應(yīng)力邊界條件，其應(yīng)力分量就能完全確定？在多連體中，要完全確定位移分量，還必須利用“位移須為單值”這個(gè)條件No，還必須考慮彈性體是否單連體In the solution of

21、elasticity problems，it is necessary to distinguish between simply connected bodies and multiply connected ones.多連體：有兩個(gè)或兩個(gè)以上連續(xù)邊界的物體，如：有孔口的物體單連體：只有一個(gè)連續(xù)邊界的物體simply connected bodies: an arbitrary closed curve lying in the body can be shrunk to a point,by continuous contraction,without passing outside it

22、s boundaries. Otherwise,the body will be said to be multiply connected.In the case of multiply connected body, there might be some arbitrary functions leading to multi-valued displacements, which are impossible in a continuous body. Then, we have to consider the condition of single-valued displaceme

23、nts to determine the stresses.In plane problems, however, we may also briefly define a simply connected body as one with only one continuous boundary and a multiply connected body as one with two or more boundaries.29 常體力情況下的簡(jiǎn)化 體力不隨坐標(biāo)而變化（重力、慣性力） 應(yīng)力分量應(yīng)滿足： （a）（b）Case of Constant Body ForcesIn many eng

24、ineering problems,the body forces are constant.On the condition of constant body forces, the compatibility equations will reduce to the homogeneous differential equation 上述方程中不含材料常數(shù)，所以對(duì)兩類平面問題都適用.Now the differential equations of equilibrium and the stress boundary conditions, as well as the compatib

25、ility equation, do not contain any elastic constant and are the same for both kinds of plane problems.只要彈性體（單連體）邊界相同，外載相同，不管是何種材料，也不管是平面應(yīng)力狀態(tài)或平面應(yīng)變狀態(tài)，應(yīng)力分布是相同的（位移及變形是否相同？）In a stress boundary problem for a simply connected body with a certain boundary and subjected to certain external forces, the stres

26、s components will have the same distribution in both plane condition.This conclusion is very helpful in the experimental analysis.（1）可將某種材料，某種狀態(tài)下所求的應(yīng)力分量的結(jié)論用于其他材料或其他狀態(tài)（邊界條件，外荷載相同）We may use any model material convenient for stress measurement instead of the structure material on which the measurement

27、 might be impossible.（a）（2）在實(shí)驗(yàn)中，可以用便于測(cè)量的材料來制造模型；或用平面應(yīng)力情況下的薄板來代替平面應(yīng)變情況下的長(zhǎng)柱體.We may use a model in plane stress condition (a thin slice) instead of one in plane strain condition (a long cylindrical body).（b）The stress components are determined by the differential equations:（a） is nonhomogeneous and, t

28、herefore, its general solution may be expressed as the sum of a particular solution and the general solution of the homogeneous system考察（a），其解由非齊次方程的特解+齊次方程的通解 特解設(shè)為：x=-Xx，y=-Yy，xy=0 （c）或 x=0，y=0，xy=-Xy-Yx或 x=-Xx-Yy，y=-Xx-Yy，xy=0只要能滿足方程即可?。╟）式求齊次方程的通解將方程變?yōu)?1)(2)滿足（1），必存在一個(gè)函數(shù)A（x，y）使得：According to diff

29、erential calculus, for (1), there exists a certain function A(x，y) so that：Rewrite同理滿足（2），必存在一個(gè)函數(shù)B（x，y）使得：So Similarly, (2) ensures the existence of another function B(x，y) so that：必存在一個(gè)函數(shù)（x,y），且Which ensures the existence of still another function (x, y) so that所以，齊次方程的通解為：We obtain the general sol

30、ution of homogeneous equations：平衡微分方程的解為：（c）Now, the superposition of the general solution with the particular solution yields the following complete solution:The function （x,y） is known as the stress function for plane problems, or the Airys stress function.（x,y）稱為平面問題的應(yīng)力函數(shù)艾瑞應(yīng)力函數(shù)With any function （

31、x,y）,the stress components so defined always satisfy the differential equations. This function （x,y）is known as the stress function or the Airys stress function. 應(yīng)力分量除滿足平衡微分方程外，還必須同時(shí)滿足相容方程，所以將（c）代入相容方程In order for the stress components to satisfy the compatibility equation as well,the stress functio

32、n must satisfy a certain equation.Or This is the compatibility equation in terms of the stress function . 2(Xx)= 2(Yy)=0 in the condition of constant body force 此方程為雙調(diào)和方程，寫為：or be simply written as：若不計(jì)體積力，即 X=0；Y=0When body forces are not considered, the solution will reduce toThus：in the solution o

33、f plane problems in terms of stresses, when the body forces are constant, it is only to solve for the stress function from the single differential equation and then find the stress components byBut these stress components must satisfy the stress boundary condition. In the case of multiply connected

34、bodies, the condition of single-valued displacements must be inspected in addition.歸納：按應(yīng)力求解平面問題時(shí)，如果體力是常量，則由 求解應(yīng)力函數(shù)，然后按求應(yīng)力分量，這些應(yīng)力分量在邊界上滿足應(yīng)力邊界條件，在多連體中，還須考慮位移單值條件.To solve the partial differential equations of elasticity together with the given boundary conditions, the direct method of solution is usua

35、lly impossible. We have to use the inverse method or the semi-inverse method.In the inverse method, some functions satisfying the differential equations are taken and examined to see what boundary conditions these functions will satisfy and thereby to know what problems they can solve. In the case of solution by Airys stress function, we select some function satisfying the compatibility equation, find the stress components, and then find the surface force components. In this way, we identify the problem which the stress function can solve.In the semi-i