2 The theory of plane problem

1、2.9 SOLUTION OF PLANE PROBLEM IN TERMS OF DISPLACEMENTFor 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 equations and boundary conditions containing the displacement

2、 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, formulate a system of differential equations and boundary co

3、nditions 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 displacement components and also some of the stress components as the

4、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.Now we proceed to formulate the differential equations and boundary conditions for solut

5、ion of a plane problem in terms of displacements.The geometric equations： 1. The differential equations The physical equations (plane stress problem)Substitution of geometric Eqs. into these equationsNow, using these relations in equilibrium equationsThe differential equations for the solution of th

6、e problem in terms of displacements. 2. Boundary conditions lx+mxy=Xmy+lxy=YStress boundary conditions of a plane problemWe obtain the stress boundary conditions of the problem in terms of displacements：To sum up,The displace components u(x,y), v(x,y) in a plane stress problem must satisfythroughout

7、 the body considered and also satisfyon the surface of the body.For a plane strain problem, it is necessary in above equations. 2.10 SOLUTION OF PLANE PROBLEM IN TERMS OF STRESSESThe two differential equations of equilibrium contain the stress components only and may be used for their solution. The

8、third differential equation can be obtained from the geometrical and physical equations by eliminating the displacement components therein.The geometrical equations of a plane problem areAdding the second derivative of x with respect to y and the second derivative of y with respect to x, we get The

9、compatibility equation for strainThe compatibility equation for strain must be satisfied by the strain components x, y and xy to ensure the existence of single-valued continuous functions u and v connected with the strain components by the geometrical equations.若所选的x、y和xy不满足这个方程，那么，由几何方程中的任意两个所求出的位移

10、分量，将不满足第三个方程例如选x=0，y=0，xy=cxy 不满足相容方程由此应变求位移第三个方程不能满足，所求u，v不存在By using physical equations, the compatibility equation can be transformed into a relation between the stress components. Substitution of the physical equations into For a plane stress problem To transform this equation into a different f

11、orm more suitable for use, we eliminate the term involving xy by using the differential equations of equilibrium. Differentiating the first equation with respect to x and the second with respect to y, adding them up and noting that xy=yx, we getSubstituting this into the compatibility equation and p

12、erforming some simplification, we obtainthe compatibility equation in terms of stresses.For a plane strain problem, an equation similar to above equation may be obtained simply byThe result is Thus,1. In the solution of a plane problem in terms of stresses, the stress components must satisfy the dif

13、ferential equations of equilibrium and the compatibility equation. Besides, they must satisfy the stress boundary conditions.2. Since the displacement boundary conditions can be expressed neither in terms of stress components nor in terms of their derivatives with respect to the coordinates, displac

14、ement boundary problems and mixed boundary problems cannot be solved in terms of stresses. In the solution of elasticity problems, it is necessary to distinguish between simply connected bodies and multiply connected ones.A body is said to be simply connected if an arbitrary closed curve lying in th

15、e body can be shrunk to a point, by continuous contraction, without passing outside its boundaries. Solid blocks and hollow spheres are examples of simply connected bodies. Otherwise, the body will be said to be multiply connected. Rings and hollow cylinders are examples of multiply connected ones.I

16、n 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 displacements to determine the stresses.In plane problems, however, we may also br

17、iefly define a simply connected body as one with only one continuous boundary and a multiply connected body as one with two or more boundaries.2.11 CASE OF CONSTANT BODY FORCES In many engineering problems, the body forces are constant, I.e., the components X and Y do not vary with coordinates throu

18、ghout the volume of the whole body.（the gravity forces , the inertia forces） On the condition of constant body forces, the compatibility equations will reduce to the homogeneous differential equation Now, the differential equations of equilibrium and the stress boundary conditions, as well as the co

19、mpatibility equations, do not contain any elastic constant and are the same for both kinds of plane problems.Hence, in a stress boundary problem for a simply connected body with a certain boundary and subjected to certain external forces, the stress components x, y, xy will be independent of the ela

20、stic properties of the body and have the same distribution in both plane stress condition and plane strain condition. This conclusion is very helpful in the experimental analysis of the stresses in a structure or its elements. （1）可将某种材料，某种状态下所求的应力分量的结论用于其他材料或其他状态（边界条件，外荷载相同） （2）We may use a model in

21、 plane stress condition (a thin slice) instead of one in plane strain condition (a long cylindrical body). （1）We may use any model material convenient for stress measurement instead of the structure material on which the measurement might be impossible.(3) Beside, in the case of a stress boundary pr

22、oblem for simply connected bodies subjected to constant body forces, a stress analysis for the action of the body forces may be converted to the analysis for the action of surface forces.The stress components x, y, xy are determined by the differential equations （a）（b）and the boundary conditionsl(x)

23、s+m(xy)s=Xm(y)s+l(xy)s=YNow, we set x=x-Xx, y=y-Yy, xy =xy and proceed to find the differential equations and boundary conditions which must be satisfied by x, y, xy.(c)Substitute them into (a), (b) and (c) and obtainl(x)s+m(xy)s=X+lXxm(y)s+l(xy)s=Y+mYyWe see that the differential equations and boun

24、dary conditions to be satisfied by x, y, xy must be the same as those in a problem with zero body forces and with surface force components X and Y increased by lXx and mYy, respectively.This conclusion suggests a process for the solution of x, y, xy: Neglect the body forces and apply fictitious surf

25、ace force components X”=lXx and Y”=mYy in addition to the original surface forces;Solve for the stress components, x, y, xy, by appropriate methods;Find x=x-Xx, y=y-Yy, xy =xy 2.12 AIRYS STRESS FUNCTION. INVERSE METHOD AND SEMI-INVERSE METHODis nonhomogeneous and, therefore, its general solution may

26、 be expressed as the sum of a particular solution and the general solution of the homogeneous systemWhen body forces are constant, the particular solution may be taken as x=-Xx，y=-Yy，xy=0 or x=0，y=0，xy=-Xy-Yxor x=-Xx-Yy，y=-Xx-Yy，xy=0Which satisfy equations(1)(2)According to differential calculus, fo

27、r (1), there exists a certain function A(x，y) so that：RewriteSimilarly, (2) ensures the existence of another function B(x，y) so that：Since xy=yx, we haveWhich ensures the existence of still another function (x, y) so thatWe obtain the general solution of homogeneous equations：Now, the superposition

28、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.In order for the stress components to satisfy the compatibility equation as well, the stress function must s

29、atisfy a certain equation.or 2(Xx)= 2(Yy)=0 in the condition of constant body force or be simply written as：When body forces are not considered, the solution will reduce toThus：in the solution of plane problems in terms of stresses, when the body forces are constant, it is only to solve for the stre

30、ss 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 bodies, the condition of single-valued displacements must be inspected in addition.To solve the partial di

31、fferential equations of elasticity together with the given boundary conditions, the direct method of solution is usually 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

32、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-inverse method, we assume the solution for the s