一种求解简支钢—混结合梁通用挠度表达式的数学解析方法

An analytical solution to general deflection of simply-supported steel-concrete composite beams

钢梁与混凝土板之间的相对滑移,会使结合梁的整体刚度降低,在静力和动力计算时,必须要考虑这种滑移带来的影响。通过直接平衡法对结合梁的基本动力理论进行推导,得到集中荷载作用下结合梁的动力响应表达式;经过一系列的数学变换,使通常只能数值求解的响应表达式满足级数求和条件,以相应的Bernoulli函数和Bernoulli数代入,即可得到集中荷载作用下简支结合梁的通用挠度表达式,此表达式不仅能体现不考虑滑移的单一材料梁的挠度结果,也能反映因滑移引起的附加挠度的影响。把理论分析结果与实测结果进行了对比分析,结果吻合良好。研究结果表明:考虑相对滑移后,结合梁的振动方程形式更为复杂,表现出与普通梁不同的动力特性;在进行结合梁的挠度计算时,本质上可把静力分析结果可看作动力响应的一种特殊形式。

The slip between a steel girder and a concrete slab may decrease the global stiffness of a composite beam,it should be taken into account in static and dynamic analyses. Here,using a derivation with the fundamental dynamic theory, an analytical solution to deflection functions of simply-supported composite beams which were conventionally solved by numerical methods was derived combined with the direct balanced method of composite beam theory. After a series of mathematical transformations, the deflection functions finally satisified series summation conditions,and then the analytical solution to deflection functions of composite beams under concentrated loads was obtained by substituting the corresponding Bernoulli functions and Bernoulli numbers. The deflection functions could not only reflect the result of common beams,but also reflect the additional deflection caused by slip. The analysis results were compared with the experimental ones,they agreed well each other. The results showed that considering relative slips in composite beams,the governing motion equations of composite beams are more complex and have different dynamic characteristics compared with those of common beams; and the static results can be taken as a special case of dynamic responses of composite beams.