Elastica of Non-Prismatic and Nonlinear Elastic Cantilever Beams under Combined Loading
<p>Geometry of the non-prismatic cantilever beam with a rectangular cross-section.</p> "> Figure 2
<p>Elastica of cantilever beam under combined loading.</p> "> Figure 3
<p>Comparisons of tip responses from this study (■) and those from the literature (●) for four degenerate cases: (<b>a</b>) prismatic nonlinear elastic beam with <math display="inline"><semantics> <mi>P</mi> </semantics></math> [<a href="#B4-applsci-09-00877" class="html-bibr">4</a>], (<b>b</b>) prismatic nonlinear elastic beam with <math display="inline"><semantics> <mi>C</mi> </semantics></math> [<a href="#B5-applsci-09-00877" class="html-bibr">5</a>], (<b>c</b>) prismatic nonlinear elastic beam with <math display="inline"><semantics> <mi>P</mi> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>W</mi> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mi>P</mi> <mo>/</mo> <mi>l</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> [<a href="#B8-applsci-09-00877" class="html-bibr">8</a>], and (<b>d</b>) non-prismatic linear elastic beam with <math display="inline"><semantics> <mi>P</mi> </semantics></math> and <math display="inline"><semantics> <mi>C</mi> </semantics></math> [<a href="#B16-applsci-09-00877" class="html-bibr">16</a>].</p> "> Figure 4
<p>Tip responses of the beam under (<b>a</b>) uniform loading, (<b>b</b>) tip point loading, and (<b>c</b>) tip couple loading.</p> "> Figure 5
<p>Elastica of beams subjected to (<b>a</b>) uniform loading, (<b>b</b>) tip point loading, (<b>c</b>) tip couple loading, and (<b>d</b>) combined loading.</p> "> Figure 6
<p>Effects of cross-sectional ratio on the (<b>a</b>) tip responses and (<b>b</b>) elastica of beams under combined loading.</p> "> Figure 7
<p>Elastica of steel, aluminum alloy, and annealed copper beams.</p> "> Figure 8
<p>Bending-moment diagrams for steel, aluminum alloy, and annealed copper beams.</p> "> Figure 9
<p>Values of (<b>a</b>) strain <math display="inline"><semantics> <mrow> <mi>ε</mi> </mrow> </semantics></math> and (<b>b</b>) stress <math display="inline"><semantics> <mrow> <mi>σ</mi> </mrow> </semantics></math> as functions of the distance <math display="inline"><semantics> <mi>H</mi> </semantics></math> from the neutral axis for steel, aluminum alloy, and annealed copper beams at clamped ends.</p> ">
Abstract
:1. Introduction
2. Geometry of the Non-Prismatic Beam
3. Mathematical Modelling
4. Solution Methodology
- 1)
- Set the input parameters of , , , , , , , and .
- 2)
- Consider a trial at the clamped end . The first trial is numbered as zero.
- 3)
- Integrate Equations (21) and (22) using the boundary conditions in Equation (24) and a trial . After completing integration, the trial solution includes in .
- 4)
- Calculate (i.e., the trial boundary condition at the tip ) for Equation (25) as follows:If , the trial solution computed in step three is the characteristic solution. If , the trial solution is the numerical solution. Here, the first convergence criterion is set to .
- 5)
- If the criterion in step four is not satisfied, increment to the previous trial to calculate the next trial as , and repeat steps 2–5.
- 6)
- Note the sign of , where and are the values of in the previous and current steps, respectively. If the sign changes, the characteristic solution of lies between and . The next trial is defined using the Regula–Falsi method, and a solution for the nonlinear equations is defined as
- 7)
- Once the numerical process is entered into the Regula–Falsi scheme, repeat step six until the following criterion is satisfied:
5. Results and Comparisons
- ■
- Steel: GPa and .0
- ■
- NP8 aluminum alloy: MPa and
- ■
- Annealed copper: MPa and
- ■
- Beam geometry: 1 m, m, m, m, and m
- ■
- Loading conditions: kN/m, kN, and kNm
- ■
- Beam geometry: , , , and
- ■
- Loading conditions: , , and for steel; , , and for the NP8 aluminum alloy; and , , and for annealed copper.
6. Conclusions
- 1)
- The tip response of increases with an increasing load along the equilibrium path. Regarding the relationship between tip response and cross-sectional ratio, the tip responses of decrease as the cross-sectional ratio increases. The tip responses in this study are in good agreement with those in the literature.
- 2)
- The mechanical properties of and were both observed to have important effects when calculating bending moments. The exponential constant , rather than the Young’s modulus , dominates elastica calculations.
- 3)
- For two loads of and , the angle parameter converges to as , , and both increase. However, for a load can exceed a value of one. The relationship between deflection and cross-sectional ratio is strongly nonlinear, as indicated by the equilibrium paths.
- 4)
- The stress values of copper and aluminum near the neutral axis are relatively larger than that of steel. The stress near the neutral axis increases as the exponent constant increases.
Author Contributions
Funding
Conflicts of Interest
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Beam Material | Mechanical Property | |||||
---|---|---|---|---|---|---|
Steel | 0.000986 | 203 | 68.0 | 0.01971 | 207,000 | 1.0 |
Aluminum alloy | 0.00481 | 149 | 67.6 | 0.09612 | 455.8 | 4.79 |
Annealed copper | 0.0657 | 130 | 52.8 | 1.314 | 458.5 | 2.16 |
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Lee, J.K.; Lee, B.K. Elastica of Non-Prismatic and Nonlinear Elastic Cantilever Beams under Combined Loading. Appl. Sci. 2019, 9, 877. https://doi.org/10.3390/app9050877
Lee JK, Lee BK. Elastica of Non-Prismatic and Nonlinear Elastic Cantilever Beams under Combined Loading. Applied Sciences. 2019; 9(5):877. https://doi.org/10.3390/app9050877
Chicago/Turabian StyleLee, Joon Kyu, and Byoung Koo Lee. 2019. "Elastica of Non-Prismatic and Nonlinear Elastic Cantilever Beams under Combined Loading" Applied Sciences 9, no. 5: 877. https://doi.org/10.3390/app9050877