用Chebyshev函数研究双模量梁变形时的解析解

Research on the analytic solution of double modulus beam deformation based on Chebyshev function

用Chebyshev函数构造双模量梁拉伸区和压缩区的轴向位移函数,然后利用双模量梁横截面剪应力公式确定了拉伸区和压缩区轴向位移函数表达式,再结合位移几何方程得到了双模量梁的弯曲微分方程和弯曲正应力公式.计算分析表明:用Chebyshev函数得到双模量梁变形时的解析解的计算精度很高,利用Chebyshev函数研究复杂载荷作用下的双模量梁弯曲变形时,可以方便得到双模量梁弯曲变形的挠曲线方程,而弹性力学方法却难以求得复杂载荷作用下双模量梁弯曲变形时的挠曲线方程.双模量梁截面的弯矩方向相反梁段的挠曲线是间断的而不是连续的,原因是两梁段弯曲时的中性轴不在同一水平线上.

Chebyshev function is used to construct the axial displacement functions of the tensile zone and the compression zone of the double modulus beam,and then the expressions of the axial displacement functions of these zones are determined by using the shear stress formula of the cross section.That combining the displacement geometric equation can obtain the bending differential equation and the bending normal stress formula of the double modulus beam.The calculation results show that when Chebyshev function is used to solve the analytic solution of double modulus beam during deformation,the calculation accuracy is very high,and when the function is used to study the bending deformation of double modulus beam under complex loads,the deflection curve equation can be obtained conveniently.However,the elastic mechanics method is difficult to obtain this deflection curve equation.The deflection curve of the beam section of the double modulus beam with opposite bending moment direction is discontinuous,whose reason is that the neutral axis of the bending of the two beam sections is not on the same horizontal line.