Effects of varying amount and location of thermal mass on dynamic heat-transfer characteristics of insulated building walls with same nominal resistance (R-n-value) are investigated numerically under steady periodic conditions using climatic data of Riyadh. Concepts of "thermal-mass energy-savings potential" (Delta) and "critical thermal-mass thickness" (L-mas,L-cr) are developed and utilized in order to determine thermal mass thickness (L-mas) required for a selected desirable percentage of energy savings. Results show that daily transmission loads are not affected by L-mas for representative days of months in summer and winter. However, for moderate months, daily cooling and heating transmission loads decrease with increasing L-mas and either diminish to zero or be reduced asymptotically to constant values. For all months, peak transmission loads and decrement factor decrease, while time lag increases, with increasing L-mas. For a given L-mas, a wall with outside insulation gives better overall performance than a wall with inside insulation. While R-n-value is constant, wall dynamic resistance (R-d-value) changes and represents actual variations in transmission loads. For Delta in the range 70-99%, L-mas.cr ranges between 6 and 30 cm by using heavyweight concrete. It is found that maximum savings in yearly cooling and heating transmission loads are about 17% and 35%, respectively, as a result of optimizing L-mas for same R-n-value. It is recommended that building walls should contain L-mas.cr that corresponds to high Delta (approximate to 95%) and with insulation placed on outside for applications with continuously operating year-round AC. (C) 2011 Elsevier Ltd. All rights reserved.