This paper is concerned with numerical solution of fluid-structure inter-action (FSI) problems involving an incompressible viscous flow and an elasticstructure. A semi-implicit partitioned method with second-order temporalaccuracy is proposed. The method separates the pressure term of the fluidequations and strongly couples it to the structure, while the remaining fluidterms and the geometrical nonlinearities are treated explicitly. A second-order projection method is used to solve the fluid equations and also as aframework for the FSI coupling. Particular attention is paid to the boundaryconditions for fluid equations and the accuracy of the fluid pressure on thecommon interface. The proposed coupling method retains the second-orderaccuracy for fully-coupled nonlinear FSI problems. Extensive numerical testsare carried out on a number of benchmark FSI problems and the second-ordertemporal accuracy for all the variables of interest (fluid velocity and pressure,and structural displacement) is demonstrated.