## Seismic Ground-Surface Strains

Seismic ground strains have been analyzed since the late 1960’s, when Newmark [366], [367], based on the “traveling-wave” assumption, proposed strain estimates to be used in the seismic design of above-ground and buried structures. Consider, e.g., a single, monochromatic wave that propagates on the ground surface with a constant velocity c without any signiﬁcant changes in its shape and without any interference from other waves. The expression for the displacement time history of such a wave can be written as:

u(x, t) = f (

t − x c

) (5.1)

The relationship between the strain, (x, t), and the (particle) velocity, v(x, t), for the wave of Eq. 5.1 along its direction of propagation can be easily shown to be:

(x, t) = ∂u(x, t) ∂x

= −1 c

∂u(x, t) ∂t

= −1 c

v(x, t) (5.2)

Equation 5.2 simply relates the space and time derivatives of the displacement ﬁeld through a proportionality constant, the inverse of the apparent propagation velocity of the waves on the ground surface. With the consideration that the displaced conﬁguration of a straight, buried pipeline is similar to that of the ground [369], [384], the maximum axial strain induced in the structure was then approximated by the maximum ground strain, max, which, from Eq. 5.2, is given by [366], [369]:

max = (vL )max c

(5.3)

where (vL )max is the maximum horizontal (particle) velocity in the longitudinal direction of the pipeline and c is the corresponding component of the apparent velocity of the waves with respect to the ground surface. The rotational ground deformation

for the evaluation of the torsional response in structures was also approximated by a similar expression. In this case, the maximum angle of rotation of the ground about the vertical axis, θmax, is estimated from Eq. 5.2 as [367], [369]:

θmax = (vT )max c

(5.4)

where (vT )max is now the maximum horizontal (particle) velocity in the transverse direction.