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1、1,Chapter 4 Concepts from Theory of Elasticity,2,Topics of Today,Chapter 4 Concepts from The Theory of Elasticity 4.1 Plane Elastic Problems 4.2 The Airy Stress Function 4.3 Prandtls Stress Function for Torsion,3,4.1 Plane Elastic Problems,4.1.1 Definition,4,4.1.1 Definition,To be classified as a pl
2、ane elastic problem, the problem must have the following characteristics:Geometry: 1) a plane body ought to consist of a region of uniform thickness t, bounded by two planes paralleling to the xy plane, and by any closed surface. 2) If the thickness t is very small compared to the dimensions in the
3、parallel planes, the problem is classified as a plane stress problem. 3) If the thickness t is very large compared to the dimensions in the parallel planes, the problem is classified as a plane strain problem.,5,4.1.1 Definition,Loading: Applied surface loads and/or body forces must be non-varying i
4、n the z direction and CAN NOT have components in the z direction. And applied loads CAN NOT exist on top and bottom surfaces. Planes stress: z = xz = yz = 0 and the stresses in the x and y directions are not functions of z. Planes strain: z = xz = yz = 0 and the strains in the x and y directions are
5、 not functions of z.A simple example illustrating the difference between a plane stress and plane strain problem is that of a narrow-beam problem versus a wide-beam problem. If a beam with a rectangular cross section is narrow, it is considered to be a plane stress problem. If a rectangular beam is
6、wide, it is considered to be plane strain.,6,4.1.2 Governing Equations,Plane stress case:,Compatibility:,Equilibrium:,7,4.1.2 Governing Equations,For xy being continuous in x and y:,Thus, it yields:,8,4.1.2 Governing Equations,For problems in which the body forces are zero: 2 (x + y) = 0,9,4.1.2 Gov
7、erning Equations,Plane strain case:,For problems with no body forces 2 (x + y) = 0,Repeating the derivation, it yields:,10,Topics of Today,Chapter 4 Concepts from The Theory of Elasticity 4.1 Plane Elastic Problems 4.2 The Airy Stress Function 4.2.1 Rectangular Coordinates 4.2.2 Polar Coordinates 4.
8、2.3 Curved Beam in Bending 4.2.4 Circular Hole in a Plate Loaded in Tension 4.2.5 Concentrated Force on a Flat Boundary 4.2.6 Disk with Opposing Concentrated Force 4.3 Prandtls Stress Function for Torsion,11,4.2 The Airy Stress Function,4.2.1 Rectangular Coordinates In this section, no body forces 2
9、 (x + y) = 0 Assume there exists a function (x, y) such that:,Equilibrium:,Equilibrium is satisfied.,12,4.2.1 Rectangular Coordinates,Compatibility:,Use the notation:,This equation is called the biharmonic equation. The function (x, y) is referred to as the Airy Stress Function.,13,4.2.1 Example 4.2
10、-1,Example 4.2-1Determine the Airy stress function for the stress field and evaluate the stress field.,14,Solution:,4.2.1 Solution to Ex. 4.2-1,B.C.s: 1. x = 0: x = xy = 0 2. x = L: 3. y = c: y = -w / b xy = 0 4. y = -c: y = xy = 0,Some observations should be made before attempting to establish the
11、Airy stress function (x, y): i) Stress function with constants and linear terms of x and y will yield zero stresses, thus only terms of xy, x2, y2 and higher orders are usable. ii) Since y is constant as a function of x on top and bottom surfaces, the stress function should not contain powers of x h
12、igher than x2.,15,iii) Since the cross section is symmetric about the y axis and y = -w/b at y = c and y = 0 at y = -c y should be an odd function of y (x, y) should only contain odd powers of y. Condition: x = L, c-cybdy = 0 is automatically satisfied. Now the Airy stress function: (x, y) = c1xy +
13、c2x2 + c3y2 + c4x2y + c5xy2 + c6x3 + c7y3 + can be reduced to:(x, y) = Axy + Bx2 + Cx2y + Dy3 + Exy3 + Fx2y3 + Gy5 Since it can NOT have x3, x4 and y2, y4 and their related terms. For 4 = 0, a y5 term is needed to respond the x2y3 term.,4.2.1 Solution (Continued),16,4.2.1 Solution (Continued),4 = 0
14、(24F + 120G)y = 0 F = -5G,Deal with xy first: x = 0 xy = 0 A + 3Ey2 = 0 A = E =0 for all y 0 y = c xy = 0 2Cx 30Gxc2 = 0 C = 15Gc2,17,4.2.1 Solution (Continued),Now, rewrite:,where B, D, and G are to be determined. y = c y = -w/b -w/b = 2B + 20Gc3 y = -c y = 0 0 = 2B + 20Gc3 = 0 Simultaneously solve
15、 these two equations to get:,18,4.2.1 Solution (Continued),19,These results are exactly same with the results in Example 2.5-3!,4.2.1 Solution (Continued),20,Topics of Today,Chapter 4 Concepts from The Theory of Elasticity 4.1 Plane Elastic Problems 4.2 The Airy Stress Function 4.2.1 Rectangular Coo
16、rdinates 4.2.2 Polar Coordinates 4.2.3 Curved Beam in Bending 4.2.4 Circular Hole in a Plate Loaded in Tension 4.2.5 Concentrated Force on a Flat Boundary 4.2.6 Disk with Opposing Concentrated Force 4.3 Prandtls Stress Function for Torsion,21,4.2.2 Polar Coordinates,The Airy stress function is written in terms of r and :,
