Question 26.14: Determine the axial force in the flanges and web of the chan......
Determine the axial force in the flanges and web of the channel section shown in Fig. P.26.14 when it is subjected to a total axial force of 40 kN. The ply properties areE_{l} =140, 000 N/mm², E_{t}=10,000 N/mm², G_{lt}=5000 N/mm², and \nu_{lt}=0.3.
The ply properties are identical to those of P.26.9 so that the ply reduced stiffnesses are also the same, i.e.,
\begin{aligned}& k_{11}=140888 N / mm ^2 \\& k_{22}=10063 N / mm ^2 \\& k_{33}=5000 N / mm ^2 \\& k_{12}=2959 N / mm ^2 \\&k_{13}=k_{23}=k_{31}=k_{32}=0\end{aligned}The transferred reduced stiffnesses are now obtained for each of the eight plies in the flanges of the section as in S.26.9 from Eqs (26.38) and (26.31). The results are presented in Table S.26.14 where the values are in kN/mm².
From Eq. (26.47),
\left\{\begin{array}{c}\sigma_x \\\sigma_y \\\tau_{x y}\end{array}\right\}=\left[\begin{array}{ccc}\bar{k}_{11} & \bar{k}_{12} & \bar{k}_{13} \\\bar{k}_{12} & \bar{k}_{22} & \bar{k}_{23} \\\bar{k}_{13} & \bar{k}_{23} & \bar{k}_{33}\end{array}\right]\left\{\begin{array}{c}\varepsilon_x \\\varepsilon_y \\\gamma_{x y}\end{array}\right\} (26.38)
A_{ ij }=\sum\limits_{ p =1}^{ N } t_{ p }\left(\bar{k}_{ ij }\right)_{ p } \quad( i =1,2,3, \ldots, j =1,2,3, \ldots) (26.47)
A_{11}=2\times0.125(140.9+44.3+44.4+10.1)=59.9\,\mathrm{kN/mm}Similarly,
\begin{aligned}& A_{22}=59.9 kN / mm \\& A_{33}=20.7 kN / mm \\& A_{12}=18.7 kN / mm \\& A_{13}=A_{23}=0\end{aligned}Then, from Eq. (26.54),
A A=\left(A_{11}A_{22}-A_{12}^{\ 2}\right)A_{33} (26.54)
A A=(59.9\times59.9-18.7^{2})20.7=67964.3Since we are only required to find the axial force in the flange then, from Eq. (26.56), it is only necessary to calculate a_{11}. From Eqs (26.55),
\begin{aligned}& a_{11}=\left(A_{22}\right)/\left(A_{11} A_{22}-A_{12}{}^2\right) \\& a_{22}=\left(A_{11}\right) /\left(A_{11}A_{22}-A_{12}{ }^2\right) \\& a_{33}=1 /A_{33} \\& a_{12}=-\left(A_{12}\right) /\left(A_{11}A_{22}-A_{12}{ }^2\right) \\& a_{13}=0 \\& a_{23}=0\end{aligned} (26.55)
E_{x}={\frac{1}{t a_{11}}} (26.56)
a_{11}=59.9\times20.7/67964.3=0.0182Then E_{x}=E_{Z}(\mathrm{flange})=1/1\times0.0182=54.9\mathrm{~kN/mm^{2}}
We now repeat the above for the web of the section. Again the ply reduced stiffnesses are those of Ex. 26.9. The transformed reduced stiffnesses for each of the four plies in the web may be calculated from Eqs (26.38) and (26.31) as before but, in this case, will be identical to those for the plies 2, 7 and 3, 6 in the flanges, since the ply angles are the same. Then, from Table S.26.14, Eq. (26.47), and noting that the ply thickness is 0.125 mm,
A_{11}=2\times0.125(44.3+44.3)=22.2\,\mathrm{kN/mm}Similarly,
\begin{aligned}A_{22} & =22.2 kN / mm \\A_{33} & =18.2 kN / mm \\A_{12} & =17.2 kN / mm \\A_{13} & =A_{23}=0\end{aligned}From Eq. (26.54),
A A=\left(22\times22.2-17.2^{2}\right)=3585.4and from Eqs (26.55),
a_{11}=22\times18.2/3585.4=0.1127Then, from Eq. (26.56),
E_{x}=E_{Z,i}(\mathrm{web})=1/0.5\times0.1127=17.7\,\mathrm{kN/mm^{2}}Therefore, for each flange,
b_{i}t_{i}E_{Z,i}=100\times1.0\times54.9=5490\,\mathrm{kN}and for the web,
b_{i}t_{i}E_{Z,i}=200\times0.5\times17.7=1770\,\mathrm{kN}Then \sum\limits_{i=1}^n b_i t_i E_{Z, i}=2 \times 5490+1770=12750 kN
so that, from Eq. (26.66),
\varepsilon_Z=\frac{P}{\sum\limits_{i=1}^n b_i t_i E_{Z, i}} (26.66)
\epsilon_{Z}={\frac{40}{12750}}=3.14\times10^{-3}Finally, from Eq. (26.63),
P_{i}=\epsilon_{Z}b_{i}t_{i}E_{x,i} (26.63)
\begin{gathered}P(\text { flanges })=3.14 \times 10^{-3} \times 5490=17.2 kN \\ P(\text { web })=3.14\times 10^{-3} \times 1770=5.6 kN\end{gathered}Check:
Total load =2\times17.2+5.6=40\,\mathrm{kN}
| Table S.26.14 | |||||||||||||||
| Ply | θ° | {\overline{{k}}}_{\mathrm{11}} | {\overline{{k}}}_{\mathrm{22}} | {\overline{{k}}}_{\mathrm{33}} | {\overline{{k}}}_{\mathrm{12}} | {\overline{{k}}}_{\mathrm{13}} | {\overline{{k}}}_{\mathrm{23}} | ||||||||
| 1,8 | 0 | 140.9 | 10.1 | 5.0 | 3.0 | 0 | 0 | ||||||||
| 2,7 | 45 | 44.3 | 44.3 | 36.3 | 34.3 | 32.7 | 32.7 | ||||||||
| 3,6 | -45 | 44.3 | 44.3 | 36.3 | 34.3 | -32.7 | -32.7 | ||||||||
| 4,5 | 90 | 10.1 | 140.9 | 5.0 | 3.0 | 0 | 0 | ||||||||