Purpose　The　purpose　of　this　paper　is　to　propose　a　new　explicit　time　integration　algorithm　for　solution　to　the　linear　and　non-linear　finite　element　equations　of　structural　dynamic　and　wave　propagation　problems.　Design/methodology/approach　The　algorithm　is　completely　explicit　so　that　no　linear　equation　system　requires　solving,　if　the　mass　matrix　of　the　finite　element　equation　is　diagonal　and　whether　the　damping　matrix　does　or　not.　The　algorithm　is　a　single-step　method　that　has　the　simple　starting　and　is　applicable　to　the　analysis　with　the　variable　time　step　size.　The　algorithm　is　second-order　accurate　and　conditionally　stable.　Its　numerical　stability,　dissipation　and　dispersion　are　analyzed　for　the　dynamic　single-degree-of-freedom　equation.　The　stability　of　the　multi-degrees-of-freedom　non-proportional　damping　system　can　be　evaluated　directly　by　the　stability　theory　on　ordinary　differential　equation.　Findings　The　performance　of　the　proposed　algorithm　is　demonstrated　by　several　numerical　examples　including　the　linear　single-degree-of-freedom　problem,　non-linear　two-degree-of-freedom　problem,　wave　propagation　problem　in　two-dimensional　layer　and　seismic　elastoplastic　analysis　of　high-rise　structure.　Originality/value　A　new　single-step　second-order　accurate　explicit　time　integration　algorithm　is　proposed　to　solve　the　linear　and　non-linear　dynamic　finite　element　equations.　The　algorithm　has　advantages　on　the　numerical　stability　and　accuracy　over　the　existing　modified　central　difference　method　and　Chung-Lee　method　though　the　theory　and　numerical　analyses.