Static and Vibrational Analysis of Partially Composite Beams Using the Weak-Form Quadrature Element Method

<table class="table-group" width="800px"><tr class="fig-image" id="a"><td><img alt="974023.fig.0011a" src="https://static.hindawi.com/articles/mpe/volume-2012/974023/figures/974023.fig.0011a.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(a) </b>Longitudinal mode shape <svg height="10.85" id="M152" style="vertical-align:-0.1254pt" version="1.1" viewbox="0 0 47.212502 10.85" width="47.212502" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.85)">
<g transform="translate(72,-63.32)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝛼</tspan>
<tspan style="font-size: 12.50px; " x="7.0391889" y="0">𝐿</tspan>
<tspan style="font-size: 12.50px; " x="19.367147" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="31.407536" y="0">2</tspan>
</text>
</g>
</g>
</svg></td></tr></table></td></tr><tr class="fig-image" id="b"><td><img alt="974023.fig.0011b" src="https://static.hindawi.com/articles/mpe/volume-2012/974023/figures/974023.fig.0011b.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(b) </b>Longitudinal mode shape <svg height="10.9125" id="M153" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 55.025002 10.9125" width="55.025002" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.9125)">
<g transform="translate(72,-63.27)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝛼</tspan>
<tspan style="font-size: 12.50px; " x="7.0391889" y="0">𝐿</tspan>
<tspan style="font-size: 12.50px; " x="19.367147" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="31.407536" y="0">3</tspan>
<tspan style="font-size: 12.50px; " x="37.659035" y="0">0</tspan>
</text>
</g>
</g>
</svg></td></tr></table></td></tr><tr class="fig-image" id="c"><td><img alt="974023.fig.0011c" src="https://static.hindawi.com/articles/mpe/volume-2012/974023/figures/974023.fig.0011c.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(c) </b>Flexural mode shape <svg height="10.85" id="M154" style="vertical-align:-0.1254pt" version="1.1" viewbox="0 0 47.212502 10.85" width="47.212502" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.85)">
<g transform="translate(72,-63.32)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝛼</tspan>
<tspan style="font-size: 12.50px; " x="7.0391889" y="0">𝐿</tspan>
<tspan style="font-size: 12.50px; " x="19.367147" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="31.407536" y="0">2</tspan>
</text>
</g>
</g>
</svg></td></tr></table></td></tr><tr class="fig-image" id="d"><td><img alt="974023.fig.0011d" src="https://static.hindawi.com/articles/mpe/volume-2012/974023/figures/974023.fig.0011d.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(d) </b>Flexural mode shape <svg height="10.9125" id="M155" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 55.025002 10.9125" width="55.025002" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.9125)">
<g transform="translate(72,-63.27)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝛼</tspan>
<tspan style="font-size: 12.50px; " x="7.0391889" y="0">𝐿</tspan>
<tspan style="font-size: 12.50px; " x="19.367147" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="31.407536" y="0">3</tspan>
<tspan style="font-size: 12.50px; " x="37.659035" y="0">0</tspan>
</text>
</g>
</g>
</svg></td></tr></table></td></tr></table>

<table>First five mode shapes of CC composite beam for <svg height="10.85" id="M150" style="vertical-align:-0.1254pt" version="1.1" viewbox="0 0 47.212502 10.85" width="47.212502" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.85)">
<g transform="translate(72,-63.32)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝛼</tspan>
<tspan style="font-size: 12.50px; " x="7.0391889" y="0">𝐿</tspan>
<tspan style="font-size: 12.50px; " x="19.367147" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="31.407536" y="0">2</tspan>
</text>
</g>
</g>
</svg> and <svg height="10.9125" id="M151" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 55.025002 10.9125" width="55.025002" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.9125)">
<g transform="translate(72,-63.27)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝛼</tspan>
<tspan style="font-size: 12.50px; " x="7.0391889" y="0">𝐿</tspan>
<tspan style="font-size: 12.50px; " x="19.367147" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="31.407536" y="0">3</tspan>
<tspan style="font-size: 12.50px; " x="37.659035" y="0">0</tspan>
</text>
</g>
</g>
</svg>.</table>

Mathematical Problems in Engineering

fig11

Figure 11

Figure 11: Static and Vibrational Analysis of Partially Composite Beams Using the Weak-Form Quadrature Element Method 