The endothelium lining human arteries is a continuum of endothelial cells. The flowing blood imposes a shear stress on the endothelium. To compute the internal stress in the endothelium, we use two alternative hypotheses: 1) The cell content is fluid-like so that at steady-state it has no shear stress. 2) The cell content is solid-like. Under hypothesis No. 1, the membrane tension in the upper cell membrane grows in the direction opposite to the blood flow at a rate equal to the blood shear stress. At the junction of two neighboring cells the membrane tension in the downstream cell is transmitted partly to the basal lamina, and partly to the upstream cell. The transmission depends on the osmotic or static pressure difference between the cell and blood. If the static pressure difference is zero, the tension in the upper cell membrane will accumulate upstream. At other values of static pressure, the cell membrane tension may increase, decrease, or fluctuate along the vessel depending on the inclination of the side walls of the cells at the junctions. To determine the sidewall inclinations, we propose to use the complementary energy theorem. Under hypothesis No. 2, the cell content can bear shear, which tends to reduce the cell membrane tension; but the cell membrane tension accumulation phenomenon discussed above remains valid. These results are used to analyze the interaction of the cell membrane and cell nucleus; and the effect of turbulences in the flow on causing large fluctuations in cell membrane tension and vertical oscillations of the nuclei. The implication of tensile stress on the permeability of the cell membrane is discussed. We conclude that for the study of mass transport and stress fibers in the endothelial cells, one should consider the interaction of neighboring endothelial cells as a continuum, and shift attention from the shear stress in the blood to the principal stresses in the cells.