Engineering (Math)]
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[Analytical and Quantitative Methods for Civil Engineering (Math)] Please finish the answer step by step and clear to se
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[Analytical and Quantitative Methods for Civil Engineering (Math)] Please finish the answer step by step and clear to se
[Analytical and Quantitative Methods for Civil
Engineering (Math)]
Please finish the answer step by step and clear to see.
Thank you!
Consider a clay layer of thickness 2d having an initial value of excess pore water pressure given by ue(2,0)=u:(2), 0 s2 s2d. The upper and the lower boundaries of the clay layer are assumed to be free-draining, the permeability of the soil adjacent to each boundary being very high compared to that of the clay as shown in Figure 4. Thus the excess pore water pressures at the ends z=0 and 2-20 are held at 0. The excess pore water pressure ue (z, t) satisfies the following 1-D consolidation equation and boundary conditions Permeable layer Vz Clay Permeable layer Figure 4 ди. а?u. = Cy 0<z<2d, t> 0 at az2 ue(0,t) = 0, ue(2d,t) = 0, t> 0 where C, is the coefficient of consolidation (a) Using the method of separation of variables (i.e. let ue(2,t) = 2 (2)T(t)), derive the following ordinary differential equations Z" + AZ = 0 T' + c T = 0 where 2 is a real constant. (2 marks) (b) Determine the fundamental solutions of the partial differential equation and boundary conditions. (11 marks) (C) Determine a formal series expansion for the excess pore water pressure ue(2, t) that also satisfies the initial condition ue(3,0) = u(2), 0 Szs 2d. (4 marks) (d) Suppose a clay layer of thickness 2d with the initial excess pore water pressure given by ue(2,0) = U (Z) = p sin oszs2d and p is a real constant. Determine the excess pore water pressure ue (z,t) in the layer. (3 marks) a 2d
Engineering (Math)]
Please finish the answer step by step and clear to see.
Thank you!
Consider a clay layer of thickness 2d having an initial value of excess pore water pressure given by ue(2,0)=u:(2), 0 s2 s2d. The upper and the lower boundaries of the clay layer are assumed to be free-draining, the permeability of the soil adjacent to each boundary being very high compared to that of the clay as shown in Figure 4. Thus the excess pore water pressures at the ends z=0 and 2-20 are held at 0. The excess pore water pressure ue (z, t) satisfies the following 1-D consolidation equation and boundary conditions Permeable layer Vz Clay Permeable layer Figure 4 ди. а?u. = Cy 0<z<2d, t> 0 at az2 ue(0,t) = 0, ue(2d,t) = 0, t> 0 where C, is the coefficient of consolidation (a) Using the method of separation of variables (i.e. let ue(2,t) = 2 (2)T(t)), derive the following ordinary differential equations Z" + AZ = 0 T' + c T = 0 where 2 is a real constant. (2 marks) (b) Determine the fundamental solutions of the partial differential equation and boundary conditions. (11 marks) (C) Determine a formal series expansion for the excess pore water pressure ue(2, t) that also satisfies the initial condition ue(3,0) = u(2), 0 Szs 2d. (4 marks) (d) Suppose a clay layer of thickness 2d with the initial excess pore water pressure given by ue(2,0) = U (Z) = p sin oszs2d and p is a real constant. Determine the excess pore water pressure ue (z,t) in the layer. (3 marks) a 2d