In this paper we present the results of Fluid-Structure Interaction (FSI) computations of an elastic solid object and a laminar incompressible viscous flow using a penalty-projection algorithm. The mathematical problem consists of the Navier-Stokes equations coupled with a suitable structure model describing the behavior of the solid element. We consider two cases for the structure model. The first model deals with the equations of incompressible solid mechanics. We propose to split this saddle-point problem into a system of decoupled elliptic equations for velocity and pressure. Since stress-free boundary conditions cannot be enforced with a pure projection method, a penalty correction term is introduced. The results demonstrate that this penalty correction term is necessary in order to improve the accuracy of the computation of boundary quantities. The second model that we consider in this work consists of a simplified shell model. This approximation has many advantages since we can simulate the mechanical behavior of thin walls without suffering from the problem of locking and also with a reduction of the number of degrees of freedom. Finallly, we show and compare the results of a series of simulations in terms of accuracy using the above described models. These cases show stability and robustness of the proposed algorithm for appropriate values of the penalty parameter together with a reduction of the computational effort compared to monolithic algorithms.