Foundation damping incorporates combined effects of energy loss from waves propagating away from a vibrating foundation (radiation damping) and hysteretic action in supporting soil (material damping). Foundation damping appears in analysis and design guidelines for force- and displacement-based analysis of seismic building response (ASCE-7, ASCE-41), typically in graphical form (without predictive equations). We derive closed-form expressions for foundation damping of a flexible-based single degree-of-freedom oscillator from first principles. The expressions are modular in that structure and foundation stiffness terms, along with radiation and hysteretic damping ratios, appear as variables. Assumptions inherent to our derivation have been employed previously, but the present results are differentiated by: (1) the modular nature of the expressions; (2) clearly articulated differences regarding alternate bases for the derivations and their effects on computed damping; and (3) completeness of the derivations. Resulting expressions indicate well-known dependencies of foundation damping on soil-to-structure stiffness ratio, structure aspect ratio, and soil damping. We recommend a preferred expression based on the relative rigor of its derivation.