Simplified procedures for finite-deformation analyses of space frames, using one beam element to model each member of the frame, are presented. Each element can undergo three-dimensional. arbitrarily large, rigid motions as well as moderately large non-rigid rotations. Each element can withstand three moments and three forces. The nonlinear bending-stretching coupling in each element is accounted for. By obtaining exact solutions to the appropriate governing differential equations, an explicit expression for the tangent-stiffness matrix of each element, valid at any stage during a wide range of finite deformations, is derived. An arc length method is used to incrementally compute the large deformation behavior of space frames. Several examples which illustrate the efficiency and simplicity of the developed procedures are presented. While the finitely deformed frame is assumed to remain elastic in the present paper, a plastic hinge method, wherein a hinge is assumed to form at an arbitrary location in the element, is presented in a companion paper.