Elastic stresses at ridges formed by (a) erosion and (b) volcanism are calc
ulated here by the method of particular integrals with a displacement-disco
ntinuity boundary element method. The two geologic scenarios require treati
ng the far-field stresses in the analyses differently. For a continuous bod
y, the elastic governing equation requires that the second partial derivati
ves of the Vertical and horizontal far-field normal stresses with respect t
o horizontal and vertical directions be defined throughout the solution dom
ain. This constrains admissible solutions for continuous bodies in both two
-and three-dimensional analyses. For example, if the horizontal far-field n
ormal stress varies linearly with depth over any depth interval in a contin
uous body, then it must be continuous and vary linearly over the entire ele
vation range of the solution domain. This far-field stress distribution per
mits the description of a ridge formed by erosion of the surrounding materi
al. A discontinuous or a piecewise linear distribution of horizontal far-fi
eld stresses does not yield admissible solutions for a continuous body but
can apply to a ridge constructed by volcanism. Stresses induced by topograp
hy in volcanic ridges will inhibit dikes from propagating to the surface, f
avor the formation of near-surface magma chambers, and promote slope instab
ility. The analyses demonstrate that topography and the extant tectonic str
esses will not by themselves fully determine the stresses in a ridge; the g
eologic history is also important.