6.3.3.2.2.1. Beam Library

The beam library calculates automatically the mass and area moment of inertia which is determined by the cross section area.

Figure 6.269 Beam property page [Library]

Library supports 8 types of cross section area such as Circular, Elliptical, Rectangular, Thin-Wall Tube, Thick-Wall Tube, Hollow Rectangular, I-Beam, and T-Beam.

• Circular

  

  \(\begin{aligned} & A=\pi {{r}^{2}} \\ & {{I}_{xx}}=0.5\pi {{r}^{4}},\text{ }{{I}_{yy}}=0.25\pi {{r}^{4}},\,\,\,\,\,\,\,{{I}_{zz}}=0.25\pi {{r}^{4}} \\ & {{A}_{sy}}={{A}_{sz}}=\frac{6(1+\nu )}{7+6\nu }\,\,\left( or\,\,9/10 \right) \\ \end{aligned}\)

  

  Figure 6.270 Properties of Circular Beam Cross Section dialog box

• Elliptical

  

  \(\begin{aligned} & A=\pi ab \\ & \,{{I}_{xx}}=\frac{\pi {{a}^{3}}{{b}^{3}}}{{{a}^{2}}+{{b}^{2}}},\text{ }{{I}_{yy}}=\frac{\pi }{4}{{a}^{3}}b,\,\,\,\,\,{{I}_{zz}}=\frac{\pi }{4}a{{b}^{3}} \\ & {{A}_{sy}}=\frac{12(1+\nu ){{a}^{2}}(3{{a}^{2}}+{{b}^{2}})}{(40+37\nu ){{a}^{4}}+(16+10\nu ){{a}^{2}}{{b}^{2}}+\nu {{b}^{4}}} \\ & {{A}_{sz}}=\frac{12(1+\nu ){{b}^{2}}(3{{b}^{2}}+{{a}^{2}})}{(40+37\nu ){{b}^{4}}+(16+10\nu ){{a}^{2}}{{b}^{2}}+\nu {{a}^{4}}} \\ \end{aligned}\)

  

  Figure 6.271 Properties of Elliptical Beam Cross Section dialog box

• Rectangular

  

  \(\begin{aligned} & A=bh \\ & {{I}_{xx}}=\frac{b{{h}^{3}}}{16}\left[ \frac{16}{3}-3.36\frac{h}{b}\left( 1-\frac{{{h}^{4}}}{12{{b}^{4}}} \right) \right](\text{ }for\text{ }b\ge h\text{ }) \\ & {{I}_{yy}}=h{{b}^{3}}/12,\,\,\,\,\,\,{{I}_{zz}}=b{{h}^{3}}/12 \\ & {{A}_{sy}}={{A}_{sz}}=\frac{10(1+\nu )}{12+11\nu },\,\,\left( or\,\,5/6 \right) \\ \end{aligned}\)

  

  Figure 6.272 Properties of Rectangular Beam Cross Section dialog box

• Thin-Wall Tube

  

  \(\begin{aligned} & A=2\pi rt \\ & {{I}_{xx}}=\frac{4{{\pi }^{2}}t{{({{R}_{o}}-0.5t)}^{4}}}{U}=\frac{4{{\pi }^{2}}t{{r}^{4}}}{U}=\frac{4{{\pi }^{2}}t{{r}^{4}}}{2\pi r}=2\pi t{{r}^{3}},\text{ }U=2\pi r \\ & {{I}_{yy}}={{I}_{zz}}=\pi {{r}^{3}}t\, \\ & {{A}_{sy}}=\,{{A}_{sz}}=\,\frac{2(1+\nu )}{4+3\nu }\,\,\left( or\,0.5 \right) \\ \end{aligned}\)

  

  Figure 6.273 Properties of Thin-Wall Tube Beam Cross Section dialog box

• Thick-Wall Tube

  

  \(\begin{aligned} & A=\pi (r_{o}^{2}-r_{i}^{2}) \\ & \,{{I}_{xx}}=0.5\pi (r_{0}^{4}-r_{i}^{4}) \\ & {{I}_{yy}}={{I}_{zz}}=0.25\pi (r_{o}^{4}-r_{i}^{4}) \\ & {{A}_{sy}}=\,{{A}_{sz}}=\frac{6(1+\nu ){{(1+{{m}^{2}})}^{2}}}{(7+6\nu ){{(1+{{m}^{2}})}^{2}}+(20+12\nu ){{m}^{2}}},\,\,\,m=\frac{{{r}_{i}}}{{{r}_{o}}} \\ \end{aligned}\)

  

  Figure 6.274 Properties of Thick-Wall Tube Beam Cross Section dialog box

• Hollow Rectangular

  

  \(\begin{aligned} & A=2(ht+{b}'{{t}_{1}})\,\,=2(ht+(b-2t){{t}_{1}})=2(ht+b{{t}_{1}}-2t{{t}_{1}})\,\, \\ & {{I}_{xx}}=\frac{2t{{t}_{1}}{{({b}'+t)}^{2}}{{(h-t_{1}^{{}})}^{2}}}{{b}'t+h{{t}_{1}}+{{t}^{2}}-t_{1}^{2}}=\frac{2t{{t}_{1}}{{(b-t)}^{2}}{{(h-t_{1}^{{}})}^{2}}}{bt+h{{t}_{1}}-{{t}^{2}}-t_{1}^{2}} \\ & {{I}_{yy}}=\frac{h{{({b}'+2t)}^{3}}-(h-2{{t}_{1}}){{{{b}'}}^{3}}}{12}=\frac{h{{b}^{3}}-(h-2{{t}_{1}}){{(b-2t)}^{3}}}{12} \\ & {{I}_{zz}}=\frac{({b}'+2t){{h}^{3}}-{b}'{{(h-2{{t}_{1}})}^{3}}}{12}=\frac{b{{h}^{3}}-(b-2t){{(h-2{{t}_{1}})}^{3}}}{12} \\ & {{A}_{sz}}=\,\,\frac{10(1+\nu ){{(1+3m)}^{2}}}{(12+72m+150{{m}^{2}}+90{{m}^{3}})+\nu (11+66m+135{{m}^{2}}+90{{m}^{3}})+10{{n}^{2}}[(3+\nu )m+3{{m}^{2}}]} \\ & m=\frac{{b}'{{t}_{1}}}{ht}=\frac{(b-2t){{t}_{1}}}{ht},\,\,n=\frac{{{b}'}}{h}=\frac{(b-2t)}{h} \\ & {{A}_{sy}}\approx \,{{A}_{sz}} \\ \end{aligned}\)

  

  Figure 6.275 Properties of Hollow Rectangular Beam Cross Section dialog box

• I-Beam

  

  \(\begin{aligned} & r=0 \\ & A=2bt{}_{f}+(h-{{t}_{f}}){{t}_{w}} \\ & {{I}_{xx}}=2{{K}_{1}}+{{K}_{2}}+2\alpha {{D}^{4}} \\ & {{K}_{1}}=bt_{f}^{3}\left[ \frac{1}{3}-0.21\frac{{{t}_{f}}}{b}\left( 1-\frac{t_{f}^{4}}{12{{b}^{4}}} \right) \right] \\ & {{K}_{2}}=\frac{1}{3}(h-{{t}_{f}})t_{w}^{3} \\ & \alpha =\frac{t}{{{t}_{1}}}\left( 0.15+0.1\frac{r}{{{t}_{f}}} \right),\text{ }\left\{ \begin{matrix} t=\min ({{t}_{w}},{{t}_{f}}) \\ {{t}_{1}}=\max ({{t}_{w}},{{t}_{f}}) \\ \end{matrix} \right. \\ & D=\frac{{{({{t}_{f}}+r)}^{2}}+r{{t}_{w}}+0.25t_{w}^{2}}{2r+{{t}_{f}}} \\ & \,{{I}_{yy}}=\frac{(h-{{t}_{f}}){{t}_{w}}^{3}+2{{t}_{f}}{{b}^{3}}}{12},\,\,{{I}_{zz}}=\frac{b{{(h+{{t}_{f}})}^{3}}-(b-{{t}_{w}}){{(h-{{t}_{f}})}^{3}}}{12} \\ & {{A}_{sz}}=\,\frac{10(1+\nu ){{(1+3m)}^{2}}}{(12+72m+150{{m}^{2}}+90{{m}^{3}})+\nu (11+66m+135{{m}^{2}}+90{{m}^{3}})+30{{n}^{2}}(m+{{m}^{2}})+5\nu {{n}^{2}}(8m+9{{m}^{2}})} \\ & m=(2n{{t}_{f}})/{{t}_{w}},\,\,n=b/h \\ & {{A}_{sy}}\approx \,{{A}_{sz}} \\ \end{aligned}\)

  

  Figure 6.276 Properties of I-Beam Cross Section dialog box

• T-Beam

  

  \(\begin{aligned} & A={{t}_{f}}b+{{t}_{w}}(h-0.5{{t}_{f}}) \\ & {{I}_{xx}}={{K}_{1}}+{{K}_{2}}+\alpha {{D}^{4}} \\ & {{K}_{1}}=bt_{f}^{3}\left[ \frac{1}{3}-0.21\frac{{{t}_{f}}}{b}\left( 1-\frac{t_{f}^{4}}{12{{b}^{4}}} \right) \right] \\ & \,{{K}_{2}}=(h-0.5{{t}_{f}})t_{w}^{3}\left[ \frac{1}{3}-0.105\frac{{{t}_{w}}}{h-0.5{{t}_{f}}}\left( 1-\frac{t_{w}^{4}}{192{{(h-0.5{{t}_{f}})}^{4}}} \right) \right] \\ & \alpha =\frac{t}{{{t}_{1}}}\left( 0.15+0.1\frac{r}{{{t}_{f}}} \right),\text{ }\left\{ \begin{matrix} t=\min ({{t}_{w}},{{t}_{f}}) \\ {{t}_{1}}=\max ({{t}_{w}},{{t}_{f}}) \\ \end{matrix} \right. \\ & D=\frac{{{({{t}_{f}}+r)}^{2}}+r{{t}_{w}}+0.25t_{w}^{2}}{2r+{{t}_{f}}} \\ & {{I}_{yy}}=\,\,\frac{{{t}_{f}}{{b}^{3}}+(h-0.5{{t}_{f}})t_{w}^{3}}{12} \\ & {{I}_{zz}}=\frac{{{t}_{w}}{{(h-0.5{{t}_{f}})}^{3}}}{12}+\,\,\frac{bt_{f}^{3}}{12}+b{{t}_{f}}{{e}^{2}} \\ & \,\,\,\,\,\,\,\,\,\,\,\,+(h-0.5{{t}_{f}}){{t}_{w}}\times {{\left[ \frac{(h+0.5{{t}_{f}}-2e)}{2} \right]}^{2}} \\ & {{A}_{sz}}=\,\frac{10(1+\nu ){{(1+4m)}^{2}}}{(12+96m+276{{m}^{2}}+192{{m}^{3}})+\nu (11+88m+248{{m}^{2}}+216{{m}^{3}})+30{{n}^{2}}(m+{{m}^{2}})+10\nu {{n}^{2}}(4m+5{{m}^{2}}+{{m}^{3}})} \\ & m=(b{{t}_{f}})/h{{t}_{w}},\,\,n=b/h \\ & {{A}_{sy}}\approx \,{{A}_{sz}} \\ \end{aligned}\)

  

  Figure 6.277 Properties of T-Beam Cross Section dialog box

• Update Cross Section Property Automatically: If this option is checked, property values are updated automatically as geometry data change without clicking Recalculate.

  [Cowper1966]

  7. 18. Cowper, The shear coefficient in Timoshenko’s beam theory. Transactions of the ASME, Journal of Applied Mechanics, JUNE, 1966. pp. 335~340.

  [Warren]

  Warren C Young. Roark’s Formulas for Stress & Strain. McGraw Hill. pp.401. 7th edition.

How to Use Beam Library

Figure 6.278 Process when using Library data

1. Select a beam cross section type among Library in Beam property page.

2. Click Properties.

3. Define Geometry Property and click Recalculate (If the user wants to calculate Shear Area Raito values when recalculating using Library data, please check Poisson ratio).

   • And then data of the area moment of inertia and the area of cross section is calculated using Library data.

   • If the user wants to input the user-defined data, input the values directly.

4. Click Close.