Analytical formulas for prismatic continuous beams of equal spans

Solving the deformation and internal force of a prismatic multispan continuous beam of equal spans is a fundamental and classic problem in the area of civil engineering. Based on the Euler–Bernoulli beam theory, this paper presents unified analytical formulas to calculate the member-end rotation and bending moment of prismatic continuous beams of equal spans. First, simple closed-form expressions to determine the beam-end rotational stiffness of an equal-span prismatic continuous beam comprising any number of spans are derived using the displacement method in structural mechanics and the auxiliary series in mathematics. Furthermore, the rotational stiffness formulas are used to derive the analytical formulas for determining the joint rotation and bending moment at the supports of continuous beams subjected to various types of static loads and actions, such as a single point load applied at mid-span, distributed load applied over the span length, and differential temperature change between the top and bottom surfaces of the beam. Moreover, the implications of the proposed formulas on some interesting academic problems are thoroughly discussed. It is observed that as the number of spans goes infinity, the beam-end rotational stiffness of an equal-span prismatic continuous beam approaches the upper limit of 2\sqrt\text3 i₀, where i₀ denotes the linear stiffness, which is the product of the modulus of elasticity (E) and the moment of inertia (I) divided by the length (l₀) of the member of single-span beams. For equal-span prismatic continuous beams with various spans, the analytical formulas of the joint rotation and bending moment at the supports have unified expressions, while the difference between formulas for different static loads and actions is solely dependent on the fixed-end bending moment of single-span beams. This set of formulas reveals the advantages of concise form, general applicability, and convenient calculation. They can reveal the influence of the number of spans on the mechanical characteristics of continuous beams and analyze real-world engineering problems, such as optimization of the launching noses for incrementally launched bridges. Additionally, the proposed formulas in this paper can serve as an important reference for course teaching in the area of structural mechanics.