We propose an experimental-theoretical engineering procedure for assessing the strength of butt-welded joints using the force (KIc) and strain (δc) criteria of the brittle and quasibrittle fracture mechanics, respectively. For this purpose, beam specimens with square and bevel welds and an initial edge crack or notch in the weld metal are tested under three-point bending. The parameter KImax controlling fracture of a bent beam with an inclined (Mode I + Mode II) crack is assessed taking into account the values of the stress intensity factors KI and KII, and the crack inclination angle θ*. We also studied the plastic zone at the crack tip and the crack propagation kinetics depending on the weld geometry and the V-notch tip radius for butt-welded joints. The data obtained allow one to rate such joints by their strength according to the fracture toughness criteria KIc and δc.