Periodic　foundations　for　seismic　isolation　are　getting　increasing　attention　in　the　field　of　civil　engineering.　Initial　stresses　due　to　the　vertical　loads　from　the　upper　structures　exist　in　periodic　foundations.　In　the　present　paper,　attenuation　zones　of　two-dimensional　periodic　foundations　including　the　effect　of　vertical　loads　are　studied　by　the　weak　form　quadrature　element　method.　Comparisons　with　existing　results　in　special　cases　without　the　effect　of　vertical　loads　are　conducted　to　validate　the　proposed　method,　and　good　agreement　is　found.　A　comprehensive　parametric　study　is　conducted　to　investigate　the　effects　of　vertical　loads　on　the　attenuation　zones.　Moreover,　frequency-domain　and　time-domain　response　analyses　of　a　two-dimensional　periodic　foundation　with　finite　unit　cells　are　conducted　to　verify　the　theoretical　results.　The　results　show　that　bound　frequencies　of　the　attenuation　zones,　corresponding　to　the　vibration　mode　of　the　core,　are　affected　by　vertical　loads.　It　is　also　found　that　vertical　loads　shift　the　complete　attenuation　zones　to　lower　frequencies　and　reduce　the　attenuation　coefficients　in　the　attenuation　zones.