Compliance mismatch between the graft and the host artery of an end-to-side (ETS) arterial bypass graft increases the intramural stress in the ETS graft-artery junction, and thus may compromise its long-term patency. The present study takes into account the effects of collagen fibers to demonstrate how their orientations alter the stresses. The stresses in a bypass graft, as a man-made bifurcation, are compared to those of its natural counterparts with different fiber orientations. The results indicate that the fiber orientation mismatch between the graft and the host artery may increase the stresses at both the heel and toe regions of the ETS anastomosis (the maximum principal stress at the heel and toe regions increased by 72 and 12%, respectively). Our observations, thus, propose that the mismatch between the collagen fiber orientations of the graft and the host artery, independent of the effect of the suture line, may induce aberrant stresses to the anastomosis of the bypass graft.