The large displacements and large rotations of three-dimensional beams are analysed by using small strain theory and unit vectors of the cross-sections without using complicated three-dimensional rotation vectors. The accurate directions of the unit vectors of the cross-sections are computed by monotonically reducing the moment constraint error by the iterative scheme. The computation at each iteration consists of two basic steps. The first step is to compute the unit vectors of the cross-sections with the previous moment constraint error, and the second step is to compute the displacements and moment constraint error by solving the linear equations of equilibrium formulated with the given unit vectors of the cross-sections. The convergence of the iterative scheme is analysed, and the accuracy and efficiency of the solution are demonstrated by the numerical examples. © 1997 John Wiley & Sons, Ltd.