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

The Solution of the Space Problemq2(1- μ)G d2w-pgdz21-2μ0xdw1-2μpg(z+ A)2(1- μ)Gdzz=h1-2μ4(1 - μ)G Pg(z + A)2 + BW :zThe stress components are:μpg(z+ A)a.=ax1-μEduJuo, =-pg(z + A)1+μ-2udxTxy =Ty =Tx =011

11 q x z z=h o 2 2 2(1 ) d 1 2 d G w g z    − = − − d 1 2 ( ) d 2(1 ) w g z A z G    − = − + − g z A B G w + + − − = − 2 ( ) 4(1 ) 1 2    ( ) 1 x y g z A      = = − + − g(z A)  z = − + = = = 0 xy yz zx    ( ) 1 1 2 x E u x       = + + −  The stress components are:

The Solution ofthe Space Problemuqpg(z + A) = α y11-μz = pg(z + A)0xTxy = Tyz = Tx = 0Z=hThe stress boundary condition:d?w=lo. + mt, +nt.VVzdz2X=lt,+mo, +ntZYfltx+mt +no, =qAI==0 =-pgA=-qpgThe displacement boundary condition:1-2μz=h = 0B =4(1 - μ)G Pg(h + Wpg12

12 q x z z=h o 2 2 2 d d w w z  = x A y g z     + = − = − ( ) 1 g(z A)  z = − +  xy =  yz =  zx = 0 The stress boundary condition: x yx zx x l m n f    + + = xy y zy y l m n f    + + = xz yz z z l m n f    + + =   z z=0 = − = − gA q g q A  = w z=h = 0 2 ( ) 4(1 ) 1 2 g q g h G B     + − − = The displacement boundary condition:

The Solution of the Space ProblemqDisplacement components:0xz=h[u=v=01-2μpg(h2 - z2) + 2q(h - W4(1 - μ)Gu(q + pgz)ox0=y1-μStress components:z =-(+ pgz)0u0VXIn soil mechanics, this equation is called1-μ00lateral pressure coefficient13

13 u = v = 0 q x z z=h o Displacement components:  ( ) 2 ( ) 4(1 ) 1 2 2 2 g h z q h z G w − + − − − =    ( ) 1 q gz x y      + − = = − (q gz)  z = − +  In soil mechanics, this equation is called 1 lateral pressure coefficient. x y z z       = = − Stress components:

The Solution of the Space ProblemS 8-3 A Semi-space Body is Subject toNormal Concentration Force at the BoundaryThe semi-spatial body, regardless of bodyforce, is subject to normal concentrationP on the horizontal boundary. The spatialaxisymmetric displacement problem issolved by:1a0u01-2u ap1a001-2μOz.00,p±0Stress boundary condition:=0014

14 §8-3 A Semi-space Body is Subject to Normal Concentration Force at the Boundary Stress boundary condition: ( ) ( ) 0, 0 0, 0 0 0 z z z z      =  =  = = The semi-spatial body, regardless of body force, is subject to normal concentration P on the horizontal boundary. The spatial axisymmetric displacement problem is solved by: 2 2 2 1 0 1 2 1 0 1 2 Z u u u z          +  − = −   +  = −  ρ

The Solution ofthe Space ProblemtheUsingSaint-venant'stake a plateprinciple, weisolated body between z=0 andz, and get the following balanceconditions:dppZF, = 0, J~ (2元pdp)o, +P= 0The answer of Boussinesg satisfying all the conditions is shown onthe following page15

15 Using the Saint-venant's principle, we take a plate isolated body between z=0 and z, and get the following balance conditions: ( ) 0 0, 2 0 F d P z z      = + =  The answer of Boussinesq satisfying all the conditions is shown on the following page. ρ dρ  z