An accurate singularity-free formulation of a three-dimensional curved Euler–Bernoulli beam with large deformations and large rotations is developed for flexible multibody dynamic analysis. Euler parameters are used to characterize orientations of cross sections of the beam, which can resolve the singularity problem caused by Euler angles. The position of the centroid line of the beam is integrated from its slope, and position vectors of nodes of beam elements are no longer used as generalized coordinates. Hence, the number of generalized coordinates for each node is minimized. Euler parameters instead of position vectors are interpolated in the current formulation, and a new C1-continuous interpolation function is developed, which can greatly reduce the number of elements. Governing equations of the beam and constraint equations are derived using Lagrange's equations for systems with constraints, which are solved by the generalized- method for resulting differential-algebraic equations (DAEs). The current formulation can be used to calculate static and dynamic problems of straight and curved Euler–Bernoulli beams under arbitrary, concentrated and distributed forces. The stiffness matrix and generalized force in the current formulation are much simpler than those in the geometrically exact beam formulation (GEBF) and absolute node coordinate formulation (ANCF), which makes it more suitable for static equilibrium problems. Numerical simulations show that the current formulation can achieve the same accuracy as the GEBF and ANCF with much fewer elements and generalized coordinates.