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摘要: 求解等跨等截面連續梁的變形和內力是土木工程領域的典型問題. 基于Euler–Bernoulli梁理論，利用位移法和輔助數列推導出任意跨數的等跨等截面連續梁梁端轉動剛度的解析表達式，進而得到連續梁支點轉角、彎矩在跨中集中荷載、滿跨均布荷載、豎向溫差作用下的通用計算公式. 研究表明：當跨數趨于無窮大時，等跨等截面連續梁的梁端轉動剛度上限值為單跨梁抗彎線剛度的
$ 2\sqrt{\text{3}} $ 倍. 不同跨數的等跨等截面連續梁可采用形式統一的解析公式計算支點轉角和彎矩，不同靜力荷載作用結果的區別僅由單跨梁的固端彎矩決定. 所得公式形式簡潔、通用性強、應用方便，能揭示跨數對連續梁力學特性的影響規律，亦可用于分析頂推施工導梁參數優化等實際工程問題.Abstract: 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{\text{3}} $ 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.-
Key words:
- continuous beams /
- rotation /
- bending moment /
- analytical formula /
- structural mechanics
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圖 1 n跨連續梁受到梁端彎矩作用的分析模型. (a) M0作用在最左端; (b) M0作用在最右端
Figure 1. Analytical model of an n-span continuous beam subjected to the beam-end moment: (a) M0 applied at the leftmost end; (b) M0 applied at the rightmost end
圖 2 n跨連續梁在第u跨受到荷載作用的分析模型
Figure 2. Analytical model of an n-span continuous beam with the uth span subjected to arbitrary loads
表 1 等跨等截面梁的計算公式
Table 1. Formulas for a prismatic continuous beam of equal spans
Case No. Analytical model Rotation, $ {z_k} $ Moment, $ {M_k} $ Parameter, $ M_1^{{\text{fix}}} $ 1 
Eq. 23 Eq. 24 $ M_1^{{\text{fix}}} = {{ - F{l_0}} \mathord{\left/ {\vphantom {{ - F{l_0}} 8}} \right. } 8} $ 2 
$ M_1^{{\text{fix}}} = {{ - ql_0^2} \mathord{\left/ {\vphantom {{ - ql_0^2} {12}}} \right. } {12}} $ 3 
$ M_1^{{\text{fix}}} = {{EI\alpha \cdot \Delta T} \mathord{\left/ {\vphantom {{EI\alpha \cdot \Delta T} h}} \right. } h} $ 4 
Eq. 25 Eq. 26 $ M_1^{{\text{fix}}} = {{ - F{l_0}} \mathord{\left/ {\vphantom {{ - F{l_0}} 8}} \right. } 8} $ 5 
$ M_1^{{\text{fix}}} = {{ - ql_0^2} \mathord{\left/ {\vphantom {{ - ql_0^2} {12}}} \right. } {12}} $ 6 
$ M_1^{{\text{fix}}} = {{EI\alpha \cdot \Delta T} \mathord{\left/ {\vphantom {{EI\alpha \cdot \Delta T} h}} \right. } h} $ Note: α and h are the linear expansion coefficient and the cross-sectional depth of the beam, respectively. The temperature changes in the top and bottom surfaces of the beam are denoted as t1 and t2, respectively. 
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