In stability theory the bifurcation of a solution is usually studied as a phenomenon depending on a parameter in the differential equation. In beam theory this bifurcation parameter is usually the conservative load. For any constant value of the bifurcation parameter it is assumed that the conditions of the Uniqueness Theorem are fulfilled. The aim of the present work is to demonstrate a case in the elementary theory of beams where the bifurcation parameter is the same as the independent variable of the differential equation, i.e. the phenomenon of bifurcation can be studied at constant loading. The investigation is based on the matrix differential equation derived in  describing the large deflections of one-dimensional elastic continua. If the bending stiffness is zero, but there are finite shear and tensile stiffnesses (this is the case if we are examining an elastic string) the solutions of the mentioned differential equation do not depend uniquely on the initial conditions, i.e. multiple equilibrium positions exist and the bifurcation belongs to the standard cusp catastrophe type.