Three dimensional contact problems of square orthotropic laminates indented by a rigid spherical indenter are solved. Simplified problems of indentations of beam and isotropic square plate are studied first to develop an efficient numerical technique and to gather the knowledge of the shape of the contact area in order to solve for the three dimensional orthotropic cases. The approach combines an exact solution method in conjunction with a simple discretization numerical scheme. Numerical sensitivity due to the ill-posed nature of the problem was experienced but was cured by enhancing the numerical approach with a least square spirit. Well agreement is obtained by comparing the results of these simplified studies with available published solutions. For isotropic plate, contact area is found to be either a circle or a hypotrochoid of four lobes featured with a shorter length of contact along the through-the- corner directions of the plate. Hertz's theory fails earlier than assuming the contact area to be a circle. In-plane dependence of the contact stress is presented to illustrate the difference of contact behavior between a square plate and a circular plate. Load-indentation relation reveals indenting a square plate is harder than indenting a circular plate of a diameter equal to the side length of the square plate. Solutions of multi-layered orthotropic cases are achieved by employing a modified analytical approach with the same numerical method. Three different configurations of plate are implemented for the orthotropic case, namely, a single layered magnesium (Mg) plate, which is slightly orthotropic, and a single and double layered plates of graphite-epoxy (G-E), which are highly orthotropic. Results for the (Mg) plate agrees with the previous isotropic case. Concept of modifying the previous hypotrochoids is introduced to seek for the contact stresses for comparatively large indentation conditions. Single-layered (G-E) plate was implemented for small indentations. The result supports the validity of Hertz's theory for small indentation and shows a relatively longer contact length in the direction of less stiffness. Two layered (G-E) plate illustrates similar distributions for the contact stresses along both of the in-plane directions with a smaller range of validity of Hertzian type behavior than the previous cases. The boundary effect prevails at the initial stage of indentation but is overcome by the effect of material orthotropy as the indentation proceeds. Thus, the contact area for small indentation appears to be the same kind of hypotrochoids as located in the isotropic case but changes to be the other type of hypotrochoids as the indentation advances. / Ph. D.