► Relations between Lmas,cr and energy savings potential (Δ) are obtained. ► Lmas,cr increases with increasing Δ. ► For given Lmas, cooling and heating energy-savings potentials are same. ► For Δ in the range 70–99%, Lmas,cr ranges between 6 and 30cm. ► Savings in loads are about 17% for cooling and 35% for heating. Effects of varying amount and location of thermal mass on dynamic heat-transfer characteristics of insulated building walls with same nominal resistance (Rn-value) are investigated numerically under steady periodic conditions using climatic data of Riyadh. Concepts of “thermal-mass energy-savings potential” (Δ) and “critical thermal-mass thickness” (Lmas,cr) are developed and utilized in order to determine thermal mass thickness (Lmas) required for a selected desirable percentage of energy savings. Results show that daily transmission loads are not affected by Lmas for representative days of months in summer and winter. However, for moderate months, daily cooling and heating transmission loads decrease with increasing Lmas 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 Lmas. For a given Lmas, a wall with outside insulation gives better overall performance than a wall with inside insulation. While Rn-value is constant, wall dynamic resistance (Rd-value) changes and represents actual variations in transmission loads. For Δ in the range 70–99%, Lmas,cr ranges between 6 and 30cm 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 Lmas for same Rn-value. It is recommended that building walls should contain Lmas,cr that corresponds to high Δ (≈95%) and with insulation placed on outside for applications with continuously operating year-round AC.