Homework20120509岩土工程分析.ppt

GEOTECHNICAL ANALYSIS Example,Example 1,Soils are often tested using a triaxial apparatus on a cylindrical soil sample. In this test, the directions of the vertical (axial) and horizontal (radial) stresses coincide with the directions of the principal stresses. The two horizontal stresses are equal (see Figure Q.1).,Figure Q.1,Assume that in a triaxial test, the following principal stresses were applied:,Note that a tensile positive notation is used here. It is assumed that the values of Youngs Modulus and Poissons ratio are 10,000 kPa and 0.3 respectively. If the soil sample being tested is still in an elastic state.,Example 1,（a）Use the elastic stress-strain relations to derive the principal strains from the given stresses. Answer：,Example 1,（b）What would be the normal and shear stresses acting on the plane that is 45 from the vertical plane? (see Figure Q.1a above) Answer：,Figure Q.1a,Example 1,（c）What would be the maximum shear stress acting on the soil sample? Also determine the plane on which the maximum shear stress is acting? Answer： max = 40 kPa when the angle is 45,Example 1,（d）Assuming that the cell pressure is kept constant (xx=yy= -40 kPa) and a cohesionless soil (c=0) with a friction angle of 30 is being tested. What is the value of the axial stress zz which will cause yielding in the sample if the soil obeys the Mohr-Coulomb yielding criterion? Answer：,Coulomb yielding criterion for purely frictional materials (tensile stress is positive):,1 = xx = -40 kPa,Hence,Example 2,A 1m diameter tunnel is located in clay at a depth of 10m below the ground surface. The tunnel is subjected to an internal pressure Pi. For simplicity this soil-structure interaction problem will be analyzed as a long, thick-walled cylinder expansion problem under undrained conditions (see Figure Q.2). Given there is no surcharge applied on the ground surface, the external pressure on the hypothetical cylinder will be assumed to be zero (i.e. Pe = 0.0 ) as shown in the figure. The undrained shear strength of the clay is known to be Cu=50 kPa and the Poissons ratio v is assumed to be 0.5 for undrained conditions.,Note: It is assumed that the tunnel is very long in the direction that is normal to the paper and therefore the problem is considered a plane strain case. As a result, the stress normal to the paper is the intermediate stress (i.e. zz = 2).,Example 2,(a) Write down the governing equation for this axisymmetric problem. Answer: Governing equation:,(b)Write down the boundary conditions for this problem. Answer: Boundary conditions:,(c)Determine the internal pressure Pi required to cause yielding of the internal boundary (r=0.5m) of the tunnel using the Tresca yield criterion.,Example 2,(c)Determine the internal pressure Pi required to cause yielding of the internal boundary (r=0.5m) of the tunnel using the Tresca yield criterion. Answer: Calculating the stresses at internal boundary (r=0.5m),where: b=20a, Pe=0, r=a,rr=-Pi ，,Hence, if the soil obeys the Tresca criterion, the soil in the internal boundary will yield when:,Pi = 49.875 kPa (or 399/8),Example 2,(d)When the actual internal pressure is only half the yielding value calculated from (c), plot the stress distributions (rr and ) in the soil surrounding the tunnel. Answer:,If Pi = half of yielding value = 24.9375 kPa,Some points,Distribution of :,Some points,Example 2,(e) Repeat question (c) but using the Von Mises yield criterion. Answer:,Example 2,A clay slope, as shown in Figure Q.3, is supported by a long retaining wall which is subjected to a horizontal force P. For the design of the retaining wall, we need to estimate the maximum load that the wall can sustain before the soil fails (this case is known as the passive loading). The undrained shear strength of the soil is known as 100kPa and the height of the retaining wall is H=10m. The soil slope angle is 45 degrees. For simplicity, the soil-wall interface is assumed to be smooth and the effect of soil weight is ignored. Use the following three methods to determine the value of force P for a unit length of the retaining wall (i.e. 1m in the direction normal to the cross section shown in Figure Q.3) at failure:,Figure Q.3,(a)The limit equilibrium method (b) The lower bound method (c)The slip line method,Example 2,(a)The limit equilibrium method Answer:,Considering the equilibrium of forces along the plane AB,F