Variation of Moments Due to the Spring Constant K in the Thermal Loaded Winkler Beam

온도하중을 받는 Winkler 보에서 스프링 계수 K값에 따른 모멘트 변화

The curvature of the Winkler beam due to the thermal loads is "$y^{{\prime}{\prime}}-{\kappa}$", where y is deflection curve, ${\kappa}$ is curvature by the thermal load. The differential equation of the Winkler beam, when loaded by thermal load, is $(EI(y^{{\prime}{\prime}}(x)-{\kappa}))^{{\prime}{\prime}}+ky(x)=0$, where k is spring constant. When the point thermal load is applied at x=a, the curvature becomes ${\kappa}={\delta}_0(x-a){\alpha}{\Delta}T/h$, where ${\alpha}$ is the coefficient of thermal expansion, ${\Delta}T$ is thermal difference between upper and lower fiber of beam, h is the depth of beam and ${\delta}_0$ is the Generalized Function. With the aid of characteristics of Generalized Functions, the solutions of the mentioned differential equations are obtained systematically. When the moment Green Function due to the point thermal load is obtained, we can get the moment, when the partial thermal load is applied, through integration of the moment Green Function within the given range. The results of this study show, when the beams loaded by point thermal load and partial thermal load, how the values of deflection and moments change depending on the spring constant k in four cases : (1)Hinge-Hinge, (2)Fix-Fix, (3)Hinge-Fix, (4)Free-Fix. When the spring constant k is zero, then the Winkler beam becomes general beam.