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

solution:

1)here stress vector along polar coordiantesystem is givenby

S=Sr.er^+Sm.em^+Sz.ez^

here polar stress has magnitude as Sr,Sm,Sz and direction as er,em,ez

here magnitude of vector is given by

Sr=cosm.Sx-sinm.Sy

Sm=sinm.Sx+cosm.Sy

where this just ives magnitudes of vector and direction is given by

r=xcosmi+ysinmj

where

er=dr/IdrI=cosm+sinm

em=dm/IdmI=-sinm+cosm

putting this together we will get vector along polar coordianteas

we get that

Sr=Sxcos^2m+Sysin^2m

Sm=Sxsin^2m+Sycos^2m

Srm=(Sy-Sx)sinm*cosm

here we have Sx=p and sy=k*P

where for k=0

Sy=0

here relation for Sm for various angle m is as follows Sm=P*sin^2m

for k=.5

Sy=.5p

Sm=Psin^2m+.5Pcos^2m

for k=1

Sy=p

Sm=psin^2m+Pcos^2m=P