The ground mass is postulated to be an infinite elastic homo

The ground mass is postulated to be an infinite elastic homogeneous isotropic medium in plane strain condition with an initial vertical stress, p, equal to the vertical ground stress at the center line of the tunnel and an initial horizontal stress, kp. For a tunnel of radius, a, derive expressions for the stress components sigma_rr, sigma_theta theta, and tau_r theta and plot the distribution of the stress sigma_theta theta as a function of theta for values of k = 0, 0.5 and 1.

Solution

Take approximation stress compressive along y axis is p, force Pa

Similarly force along x axis is kP.a

tRANSFORM TO r,THETA coordinates by transformation using rotation angles

{ Srr    Sr,theta             ]

[S r theta Stheta,theta ]    =     { ctheta     S theta} {Sxx    Sxy}   { Ctheta   -S theta

                                                     { -stheta   ctheta}    {Sxy     Syy} { S theta     C theta}

obtain Srr = Kp cos² theta + p sin² theta

S _{theta theta} = Kp sin² theta + p cos² theta

S _{theta, r} = p sin theta cos theta -Kp sintheta cos theta

for K=1 S theta ,theta = p

for K=0 S theta theta =p cos² theta

etc