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Bending Deflection of a Cantilever Beam Subjected to Axial Pressure and Transverse Force
DOI: 10.12677/IJM.2022.111003, PDF, HTML, XML, 下载: 72  浏览: 526

Abstract: The small bending deflection and large bending deflection of a cantilever beam with constant cross section subjected to axial pressure and transverse force are studied, respectively. The analytical function of small deflection curve of the beam is deduced. A mathematical model and a calculation method of large deflection curve of the beam are established. Finally, two calculation examples are given.

1. 引言

2. 悬臂梁在轴向压力和横向力联合作用下的小变形弯曲

Figure 1. A cantilever beam subjected to axial pressure and transverse force

$M={F}_{2}\left(l-x\right)+\underset{_}{{F}_{1}\left[v\left(l\right)-v\left(x\right)\right]}$ (1)

Figure 2. Free-body diagram of the beam from the cross section of abscissa x to the free end in the bending deformation position

$EI\text{ }{v}^{″}\left(x\right)={F}_{2}\left(l-x\right)+\underset{_}{{F}_{1}\left[v\left(l\right)-v\left(x\right)\right]}$ (2)

$v\left(0\right)=0$${v}^{\prime }\left(0\right)=0$ (3)

$v\left(x\right)=\frac{{F}_{2}}{{F}_{1}}\sqrt{\frac{EI}{{F}_{1}}}\left\{\mathrm{tan}\left(l\sqrt{\frac{{F}_{1}}{EI}}\right)\left[1-\mathrm{cos}\left(x\sqrt{\frac{{F}_{1}}{EI}}\right)\right]+\mathrm{sin}\left(x\sqrt{\frac{{F}_{1}}{EI}}\right)\right\}-\frac{{F}_{2}}{{F}_{1}}x$ (4)

3. 悬臂梁在轴向压力和横向力联合作用下的大变形弯曲

${F}_{\text{S}}=-{F}_{1}\mathrm{sin}\theta \left(s\right)-{F}_{2}\mathrm{cos}\theta \left(s\right)$ (5)

Figure 3. Free-body diagram of the beam from the cross section of arc coordinate s to the free end in the bending deformation position

${F}_{\text{S}}=\frac{\text{d}M}{\text{d}s}$ (6)

$M=EI\kappa =EI{\theta }^{\prime }\left(s\right)$ (7)

$EI{\theta }^{″}\left(s\right)=-{F}_{1}\mathrm{sin}\theta \left(s\right)-{F}_{2}\mathrm{cos}\theta \left(s\right)$ (8)

$\theta \left(0\right)=0$${\theta }^{\prime }\left(l\right)=0$ (9)

$x\left(s\right)={\int }_{0}^{s}\mathrm{cos}\theta \left(\xi \right)\text{d}\xi$ (10)

$y\left(s\right)={\int }_{0}^{s}\mathrm{sin}\theta \left(\xi \right)\text{d}\xi$ (11)

4. 算例

Figure 4. Small deflection curve of the beam

Figure 5. Large deflection curve of the beam

5. 结束语

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