《弹性力学》课程教学课件（英文讲稿）Chapter 8 The Solution of the Space Problem

ElasticityChheuoeaem

1 Elasticity

The Solution ofthe Space ProblemChapter 8 The Solution of the Space Problem$8-1Solving the SpaceProblemAccording totheDisplacement$8-2The Semi-space Body Subjected to Gravity andUniformlyDistributedPressure88-3A Semi-space Body is Subject to NormalConcentration Force at the Boundaryg8-4 Solving the Space Problem According to the Stressg8-5The Torsion of Equal-Section Straight Pole88-6Membrane Analogy of theTorsion Problem88-7 The Torsion of Elliptic Section Pole$8-8 The Torsion of Rectangular Section Pole2

2 Chapter 8 The Solution of the Space Problem §8-5 The Torsion of Equal-Section Straight Pole §8-7 The Torsion of Elliptic Section Pole §8-6 Membrane Analogy of the Torsion Problem §8-8 The Torsion of Rectangular Section Pole §8-1 Solving the Space Problem According to the Displacement §8-2 The Semi-space Body Subjected to Gravity and Uniformly Distributed Pressure §8-3 A Semi-space Body is Subject to Normal Concentration Force at the Boundary §8-4 Solving the Space Problem According to the Stress

The Solutionof the Space ProblemMaterial mechanics has solved the torsion problems of roundsection pole, but it can't be used to analyze the torsion problemsof non-round section pole. For the torsion of arbitrary sectionpole, it is a relatively simple spatial problem. According to thecharacteristic of the problem, this chapter first gives thedifferential functions and boundary conditions, which the stressfunction should satisfy to solve the torsion problems. Then, inorder to solve the torsion problems of relatively complex sectionpole, we introduce the method of membrane analogy.3

3 Material mechanics has solved the torsion problems of round section pole, but it can’t be used to analyze the torsion problems of non-round section pole. For the torsion of arbitrary section pole, it is a relatively simple spatial problem. According to the characteristic of the problem, this chapter first gives the differential functions and boundary conditions, which the stress function should satisfy to solve the torsion problems. Then, in order to solve the torsion problems of relatively complex section pole, we introduce the method of membrane analogy

The Solution of the Space ProblemS 8-1 Solving the SpaceProblem Accordingtothe DisplacementThe strain component is represented by the deformationcomponent, and the physical equation can be rewritten as:EowEOuu72(1 + μ)aydx1+u1-2μEuOvEduH2(1 + μ)Ozax1 +a1u2AEavauEwuA2(1 +yaxOzu1+μ-2uEμOWhere 0=&.+&. +c.X1+ μ-2u1一4

4 §8-1 Solving the Space Problem According to the Displacement The strain component is represented by the deformation component, and the physical equation can be rewritten as: Where x y z     = + + ( ) 1 1 2 ( ) 1 1 2 ( ) 1 1 2 x y z E u x E v y E w z                 = + + −   = + + −   = + + −  ( ) 2(1 ) ( ) 2(1 ) ( ) 2(1 ) yz zx xy E w v y z E u w z x E v u x y         = + +     = + +     = + +   ( ) 1 1 2 x x E       = + + −

The Solution of the Space ProblemBy substituting the above equation into the equilibrium differentialequation, we get:E00atyx1agatruX=0f=0Ozaxay2(1+ μ) (1- 2μ axao,atyaty+f=0E100azayax+Vv=(2(1+μ) (1-2 μ yOtyao.atxf=0OzaxayE1a0VW2(1+ μ) (1- 2 μ oza2a2a?72+Qy2ax?az2The above equation is the basic differential equation for solvingspace problems by displacement.5

5 By substituting the above equation into the equilibrium differential equation, we get: ( ) 1 2 0 2 1 1 2 x E u f x       +  + =   + −    ( ) 1 2 0 2 1 1 2 y E v f y       +  + =   + −    ( ) 1 2 0 2 1 1 2 z E w f z       +  + =   + −    2 2 2 2 2 2 2 x y z  +   +    = The above equation is the basic differential equation for solving space problems by displacement. 0 0 0 x zx yx x y zy xy y z xz yz z f x y z f y z x f z x y              + + + =         + + + =          + + + =    